Color Image Scrambling Technique Based on Transposition of ...
16
Research Article Color Image Scrambling Technique Based on Transposition of Pixels between RGB Channels Using Knight’s Moving Rules and Digital Chaotic Map Adrian-Viorel Diaconu, 1,2 Alexandru Costea, 3 and Marius-Aurel Costea 1 1 IT&C Department, Lumina—e University of South-East Europe, 021187 Bucharest, Romania 2 ETTI Faculty, University Politehnica of Bucharest, 061071 Bucharest, Romania 3 Faculty of Electronic and Information Military Systems, Military Technical Academy, 050141 Bucharest, Romania Correspondence should be addressed to Adrian-Viorel Diaconu; [email protected] Received 20 November 2013; Accepted 5 March 2014; Published 24 April 2014 Academic Editor: Teh-Lu Liao Copyright © 2014 Adrian-Viorel Diaconu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nowadays, increasingly, it seems that the use of rule sets of the most popular games, particularly in new images’ encryption algorithms designing branch, leads to the crystallization of a new paradigm in the field of cryptography. us, motivated by this, the present paper aims to study a newly designed digital image scrambler (as part of the two fundamental techniques used to encrypt a block of pixels, i.e., the permutation stage) that uses knight’s moving rules (i.e., from the game of chess), in conjunction with a chaos-based pseudorandom bit generator, abbreviated PRBG, in order to transpose original image’s pixels between RGB channels. eoretical and practical arguments, rounded by good numerical results on scrambler’s performances analysis (i.e., under various investigation methods, including visual inspection, adjacent pixels’ correlation coefficients’ computation, key’s space and sensitivity assessment, etc.) confirm viability of the proposed method (i.e., it ensures the coveted confusion factor) recommending its usage within cryptographic applications. 1. Introduction It is well known that, clearly, images are considered to contain a huge amount of information (e.g., a family photos might tell not only who are its members but also their rough ages or physical features, etc.) and thus, as an essential and integrated part of the advanced data protection techniques (i.e., image encryption [1–5] and watermarking [6–12]), digital image scramblers are designed to transform clear images into unin- telligible ones (i.e., whose inherent information is protected from any unauthorized use). In recent years, besides classical approaches in the design- ing of bidimensional bijection based digital image scramblers (e.g., chaos-based [13–17], cellular automata based [18, 19], interpixel displacement or image-blocks transposition based [20–22], spatial transform based [23, 24], and matrix decom- position based [25] scramblers), few new designs based on rule sets of the most popular games (e.g., Sudoku puzzle based [26–28], Chinese chess knight’s tour based [29, 30], Rubik’s cube principle based [31, 32], and Poker shuffling rules based [33]) found their rightful place. is paper aims to contribute to the crystallization of reminded paradigm by presenting a new model of digital image scrambler, based on transposition of pixels between RGB channels using knight’s moving rules (i.e., from the game of chess) in conjunction with a chaos-based PRBG. e rest of this paper is organized as follows. Section 2 presents, as comprehensively possible, the design of proposed digital image scrambler. Section 3 showcases scrambler’s per- formance analysis. Finally, Section 4 concludes the work carried out. 2. Algorithm Description/Methodology is section deals with the presentation, as comprehensively possible, of the designing stages of the proposed digital image scrambler (i.e., use of knight’s moving rules, use of digital Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 932875, 15 pages http://dx.doi.org/10.1155/2014/932875
Transcript of Color Image Scrambling Technique Based on Transposition of ...
Research ArticleColor Image Scrambling Technique Based on Transposition ofPixels between RGB Channels Using Knightrsquos Moving Rules andDigital Chaotic Map
Adrian-Viorel Diaconu12 Alexandru Costea3 and Marius-Aurel Costea1
1 ITampC Department LuminamdashThe University of South-East Europe 021187 Bucharest Romania2 ETTI Faculty University Politehnica of Bucharest 061071 Bucharest Romania3 Faculty of Electronic and Information Military Systems Military Technical Academy 050141 Bucharest Romania
Correspondence should be addressed to Adrian-Viorel Diaconu adriandiaconuluminaorg
Received 20 November 2013 Accepted 5 March 2014 Published 24 April 2014
Academic Editor Teh-Lu Liao
Copyright copy 2014 Adrian-Viorel Diaconu et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Nowadays increasingly it seems that the use of rule sets of the most popular games particularly in new imagesrsquo encryptionalgorithms designing branch leads to the crystallization of a new paradigm in the field of cryptographyThus motivated by this thepresent paper aims to study a newly designed digital image scrambler (as part of the two fundamental techniques used to encrypta block of pixels ie the permutation stage) that uses knightrsquos moving rules (ie from the game of chess) in conjunction with achaos-based pseudorandom bit generator abbreviated PRBG in order to transpose original imagersquos pixels between RGB channelsTheoretical and practical arguments rounded by good numerical results on scramblerrsquos performances analysis (ie under variousinvestigationmethods including visual inspection adjacent pixelsrsquo correlation coefficientsrsquo computation keyrsquos space and sensitivityassessment etc) confirm viability of the proposed method (ie it ensures the coveted confusion factor) recommending its usagewithin cryptographic applications
1 Introduction
It is well known that clearly images are considered to containa huge amount of information (eg a family photos mighttell not only who are its members but also their rough ages orphysical features etc) and thus as an essential and integratedpart of the advanced data protection techniques (ie imageencryption [1ndash5] and watermarking [6ndash12]) digital imagescramblers are designed to transform clear images into unin-telligible ones (ie whose inherent information is protectedfrom any unauthorized use)
In recent years besides classical approaches in the design-ing of bidimensional bijection based digital image scramblers(eg chaos-based [13ndash17] cellular automata based [18 19]interpixel displacement or image-blocks transposition based[20ndash22] spatial transform based [23 24] and matrix decom-position based [25] scramblers) few new designs based onrule sets of themost popular games (eg Sudokupuzzle based[26ndash28] Chinese chess knightrsquos tour based [29 30] Rubikrsquos
cube principle based [31 32] and Poker shuffling rules based[33]) found their rightful place
This paper aims to contribute to the crystallization ofreminded paradigm by presenting a new model of digitalimage scrambler based on transposition of pixels betweenRGB channels using knightrsquos moving rules (ie from thegame of chess) in conjunction with a chaos-based PRBG
The rest of this paper is organized as follows Section 2presents as comprehensively possible the design of proposeddigital image scrambler Section 3 showcases scramblerrsquos per-formance analysis Finally Section 4 concludes the workcarried out
2 Algorithm DescriptionMethodology
This section deals with the presentation as comprehensivelypossible of the designing stages of the proposed digital imagescrambler (ie use of knightrsquos moving rules use of digital
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 932875 15 pageshttpdxdoiorg1011552014932875
2 Mathematical Problems in Engineering
(a)
R0
R1
R2
R3R4
R5
R6
R7
(b)
I0(i minus 2 j minus 1) I0(i minus 2 j + 1)
I0(i minus 1 j + 2)
I0(i + 1 j + 2)
I0(i + 2 j + 1)I0(i + 2 j minus 1)
I0(i + 1 j minus 2)
I0(i minus 1 j minus 2)
I0(i j)
(c)
Figure 1 Knightrsquos moving rules (a) Knightrsquos possible moves (b) movesrsquo possible annotations (ie as rules) and (c) rulesrsquo associatedcoordinates
chaotic map and description of the main procedures andalgorithms)
21 Use of Knightrsquos Moving Rules in Scramblerrsquos DesignKnightrsquos moving rules unlike those of all other pieces followan ldquoelbowedrdquo nonlinear shape (ie in formof letter 119871 in otherwords a horizontal displacement of two squares followed bya vertical displacement of one square or a vertical displace-ment of two squares followed by a horizontal displacement ofone square in either direction)Thus as shown in Figure 1(a)one can define a set of eight possiblemoves each of which canbe labeled as suggested in Figure 1(b)
Considering any 24-bit color bitmap 1198680 of the size 119898 times
119898 times 119896 where 119896 isin RGB represents imagersquos color channelas a chessboard with fixed orientation and any pixel 119868
0(119894 119895 119896)
within this image as knightrsquos initial position each movingrule (ie pixelrsquos new destination) can be mathematicallyexpressed in relative coordinates as shown in Figure 1(c)
Since a digital image scrambler aims among others tominimize adjacent pixelsrsquo correlation we have considered thefollowing slightly different approaches
(i) in order to ensure the fact that moved pixelsrsquo valueswill be less correlated with those of their new neigh-boring pixels any pixel 119868
0(119894 119895 119896) will not be moved
once but 119899 times using the same moving rule 119877119909
(ii) reduced susceptibility of correlation between movedpixelrsquos value and of new neighboring pixelsrsquo (ieachieved by applying previous approach) will bestrengthened by the fact that once pixelrsquos new coordi-nates (ie 119894 and 119895) have been determined accordingto a binary random variable 119875
119910119911 its value will be
simultaneously transposed between color channels
Obviously a series of further questions arises for exampleldquowhich are 119899rsquos limits of variationsrdquo and ldquowhat happens ifpixelrsquos new destinationrsquos coordinates exceed imagersquos limitsrdquoor ldquoin what way the random binary variable 119875
119910119911influences
pixelsrsquo transposition between imagersquos color channelsrdquoThe answers to these questions (without any restriction
ie one can think of other solutions) are as follows
(i) from a theoretical perspective 119899 can accommodateany positive integer value however in practical terms(related to PRBGrsquos use as it will be discussed furtherwithin this section) its value will be restricted to an8-bit maximum value (ie 0 le 119899 le 255)
(ii) indeed there is an imminent possibility that pixelsrsquonew destinationrsquos coordinates exceed imagersquos limits(especially when pixels to be moved are situated atimagersquos extreme limits or when 119899rsquos value is close tomaximum) and therefore considering the image asbeing virtually cyclic 119894 and 119895 coordinates are to bekept within [1 119898] interval (ie in a relatively plasticbut more proper expression ldquoexitingrdquo through oneside of the image the pixel will ldquoreturnrdquo within theimage on the other side completing ldquoin a natural wayrdquomotionrsquos specific shape)
(iii) when it comes to transpose pixelsrsquo values betweenimagersquos color channels the basic idea is to ensure thatall pixels are removed from their initial color plan(eg considering any pixel 119868
0(119894 119895R) ie belonging
to the red plan its destination color plane will beforced to green or blue ie 119896 isin GB) Thiscriterion is satisfied by applying for example (ienotwithstanding the fact that other rules may beproposed) transposing rules presented in Figure 2
22 Use of the Chaotic Map in Scramblerrsquos Design Althoughany PRBG can be used for example [34ndash40] due to its goodcryptographic properties and suitability for cryptographicapplication that is large key space and good randomnessfeatures as proven in [41] PRNG model (1) was usedwithin designing and testing stages of this work On PRNGrsquosoutput sequence of real numbers themultilevel discretizationmethod [42] (eg with four thresholds ie 2-bit encoding ofeach interval) was applied resulted dibits being spread intotwo different files (ie ldquoBits Atxtrdquo containing dibitrsquos first bitand resp ldquoBits Btxtrdquo containing dibitrsquos second bit)
Using random sequences of real numbers generated bythe orbits of 119891
119879(1) designed based on a binary composition
Mathematical Problems in Engineering 3
G BRGBR
GB RGBR
0 1
0
1
Pz
Py
(a)
G BR GBR
GBR GBR
0 1
0
1
Pz
Py
(b)
0 1
0
1
GBR
GBR
GBR
Pz
Py
GB R
(c)
Figure 2 Correlation between the random binary variable 119875119910119911
and pixelsrsquo transposing rules when working on pixelsrsquo from (a) red colorchannel (b) blue color channel and (c) green color channel
Require m k imagersquos dimensionsEnsure 119888 a temporary counter initially set to 0
KMR a matrix of the size119898 sdot 119898 sdot 119896 initially set to 0 (ie KMR = zeros (119898119898 119896))RNS a matrix of the size119898 sdot 119898 sdot 119896 initially set to 0 (ie RNS = zeros (119898119898 119896))
for 119901 =1 kfor 119902 =1 m
for 119903 = 1 n119861119910119905119890119860= 119904119905119903119888119886119905 (Bits Atxt (119888 + 119904)) 119904 = 1 8 takes 8 bits from file Bits A
119861119910119905119890119861= 119904119905119903119888119886119905 (Bits Btxt (119888 + 119904)) 119904 = 1 8 takes 8 bits from file Bits B
KMR (119894 119895 119896) = 1198871198941198992119889119890119888 (119861119910119905119890119860) update KMR matrix
RNS (119894 119895 119896) = 1198871198941198992119889119890119888 (119861119910119905119890119861) update RNS matrix
119888 = 119888 + 8 update the temporary counterend
endend
Procedure 1 Computing KMR and RNS (Bits A Bits Bm k)
(2) of two identical one-dimensional chaotic discrete dynam-ical systems of form (22) in conjunction with the method ofdiscretization previously referenced 119898 sdot 119898 sdot 119896 sdot 8 dibit pairshave been generated (ie 6291456 bits were written in eachfile) this number being as seen directly proportional to theimage dimensions Consider
119891119879= 1198911(1199091
119894 1199031) lowast 1198912(1199092
119894 1199032) =
1198911(1199091
119894 1199031) + 1198912(1199092
119894 1199032)
1 minus 1198911(1199091
119894 1199031) sdot 1198912(1199092
119894 1199032)
(1)
where 11990910 11990920are the initial conditions 119903
1 1199032are the control
parameters and 1199091
119894 1199092119894are the two orbits obtained by
recurrences 1199091119894+1
= 1198911(1199091
119894 1199031) and 1199092
119894+1= 1198912(1199092
119894 1199032) for any
119894 isin 0 1 2 Consider
119886 lowast 119887 =119886 + 119887
1 minus 119886 sdot 119887 (2)
1198911 [minus1 1] 997888rarr [minus1 1]
1198911(119909 1199031) =
2
120587arctg (ctg (119903
1sdot 119909)) 119903
1isin [1 10]
1198912 [minus1 1] 997888rarr [minus1 1]
1198912(119909 1199032) =
2
120587arctg (ctg (119903
2sdot 119909)) 119903
2isin [1 10]
(3)
In our scramblerrsquos design and during subsequent tests119891119879rsquos initial seeding points respectively control parametersrsquo
values were chosen as follows 11990910= 0687754925117 and
1199031= 5938725025421 respectively 1199092
0= minus0013462335467
and 1199032= 1237490188615
Under the previous circumstances two matrices werecomputed hereafter referred to as KMR and RNS (ie theone associated with knightrsquos moving rules and respectivelythe one associated with the number of steps to be appliedon each rule) Procedure 1 describes how KMR and RNS arecomputed
KMR and RNS matrices are used to establish values of119877119909 119875119910119911 and 119899 as Figure 3 suggests It can be noticed that
we are dealing with a two-step algorithm Thus accordingto 119875119910119911rsquos value during the first step pixel 119868
0(119894 119895 119896) is moved
119899119886times using knightrsquos rule 119877
119886and then transposed to other
color planes while during the second step with its new
4 Mathematical Problems in Engineering
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
X X X X X X X X
b0 b1 b2 b3 b4 b5 b6 b7
b0b1b2b3b4b5b6b7
nb
na
RNS(i j)
(b)
Figure 3 Method of determining 119877119909 119875119910119911 and 119899 values (a) extraction of 119877
119909and 119875
119910119911values and (b) establishing the number of steps for each
rule
Require 119894 119895 119896 current pixelrsquos (ie the one to be moved) coordinates119870Rule = 1198891198901198882119887119894119899(KMR(119894 119895 119896)119877119886= 119870Rule (1 3) 119877
119886= 1198871198941198992119889119890119888 (119877
119886) compute Knightrsquos rule 119877
119886
119877119887= 119870Rule (5 7) 119877
119887= 1198871198941198992119889119890119888 (119877
119887) compute Knightrsquos rule 119877
119887
119875119910= 119870Rule (4) 119875
119911= 119870Rule (8) compute 119875
119910119911binary variable used to establish transposition rule
119899119886= RNS(119894 119895 119896) number of steps to be applied using rule 119877
119886
119899119887= 1198891198901198882119887119894119899 (119899
119886)
119899119887= 119891119897119894119901119897119903 (119899
119887)
119899119887= 1198871198941198992119889119890119888 (119899
119887) number of steps to be applied using rule 119877
119887
Procedure 2 Extracting features(KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
coordinates the pixel is moved 119899119887times using knightrsquos rule
119877119887and then into the another color planeProcedure 2 shows how 119875
119910119911 119877119886 119899119886 119877119887 and 119899
119887are
computed for each imagersquos pixel
23 Practical Example In order to facilitate understandingof those previously described the example shown in Figure 4comes into aid of the reader For this example a colorbitmap of the size 12 times 12 times 3 a starting pixel 119868
0(3 6 1)
119875119910119911
= 0 0 1198773as first stage rule 119877
5as second stage
rule usage of 1198773three times and usage of 119877
5five times are
considered
24 Other Useful Procedures Before starting with thedescription of Algorithms 1 and 2 two more procedures mustbe properly described namely the one which deals withcomputation of pixelsrsquo new coordinates both for scramblingand descrambling algorithms
Thus with 1198680representing the pixelsrsquo values matrix of a
24-bit color bitmap of the size 119898 times 119898 and 119899 representingnumber of steps to be appliedwith rule119877
119909 during scrambling
process pixelsrsquo new coordinates can be computed usingProcedure 3
On the basis thatmoving rules are symmetric (eg effectsof rule 119877
3are reversed by applying rule 119877
7) during descram-
bling process pixelsrsquo new coordinates can be computed usingthe same procedure but after modifying one of the functioncallrsquos parameters as Procedure 4 suggests
25 Scrambling and Descrambling Algorithmsrsquo DescriptionDepending on random binary variable 119875
119910119911value pixelsrsquo
color values are transposed between RGB channels usingProcedure 5 (ie during imagersquos scrambling process) andrespectively Procedure 6 (ie during imagersquos descramblingprocess)
3 Analysis and Comparison Results
As a general requirement for any digital image scrambler theoutput image should be greatly different in comparison withits plain version (ie from statistical point of view) To quan-tify this requirement in addition to visual assessment fewstatistical estimators can be used the most commonly onesbeing presented in the following subsections and accord-ing to an already widely used conventional methodology[43 44]
31 Visual Analysis The purpose of visual testing is tohighlight presence of similarities between plain image and itsscrambled version (ie if the scrambled image does or doesnot contain any features of the plain image) For the proposedscrambler same as for all digital image scramblers expecta-tions are to rearrange plain imagersquos pixels in a deterministicway but with a random-like appearance
Visual testing was performed on the 512 times 512 pixels24-bit Lena Peppers and Baboon color bitmaps from theUSC-SIPI miscellaneous image dataset [45] Figure 5 depicts
Mathematical Problems in Engineering 5
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 = 1 k
for 119902 = 1 mfor 119903 = 1 m
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119886
0(119894 119895 119896) = Compute limits on scrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
119868119887
0(119894 119895 119896) = Compute limits on scrambling (119868119886
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
Transpose pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 1 Scrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 =k minus1 1
for 119902 =m minus1 1for 119903 =m minus1 1
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119887
0(119894 119895 119896) = Compute limits on descrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
119868119886
0(119894 119895 119896) = Compute limits on descrambling (119868119887
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
Transpose pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 2 Descrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
these test plain images whereas their scrambled versionsare showcased in Figure 6 By comparing them one can saythat there is no perceptual similarity (ie no visual infor-mation can be observed in the processed versions of plainimages)
32 Adjacent Pixelsrsquo Correlation Coefficientsrsquo Analysis It iswell known the fact that generally in plain images anyarbitrarily chosen pixels are strongly correlated with theiradjacent ones (either they are diagonally vertically or hor-izontally oriented) [5 17 31] With respect to this idea an
efficient digital image scrambling schememustminimize thiscorrelation as much as possible
Whilst Figure 7 showcases the correlation distributionsof horizontally (a) vertically (b) and diagonally (c)adjacent pixels for the Baboon plain image Figure 8showcases the correlation distributions (ie for thesame pixelsrsquo adjacency cases) for the Baboon scrambledimage
At the same time all APCCs (computed over 10000 pairsof adjacent pixels randomly selected for each of the testingdirections and for each of the color channels) are summarizedin Table 1 for each of the test images
6 Mathematical Problems in Engineering
1
2
3
(a)
1 2 3
4
5
(b)
(c)
Figure 4 Pixel scramblingtransposition example (a) First stagersquos output image (b) second stagersquos output image and (c) final output image
(a) (b) (c)
Figure 5 Test plain images used during algorithmrsquos testing procedures (a) Lena (b) Peppers and (c) Baboon
It can be easily noticed that neighboring pixels in theplain images are highly correlated that is APCCsrsquo valuesare too high very close to one On the contrary in cases ofscrambled images those values are close to zero meaningthat all neighboring pixels considered in tests are weaklycorrelated which is the expected result [5 46 47]
33 Other Qualitative Measurementsrsquo Analysis Whereasthrough the visual assessment and APCCsrsquo analysis good
scrambling effects are highlighted based on MSE (meansquared error) NPCR (number of pixel change rate) andUACI (unified average changing intensity) measures a betterandmore objective assessment of the proposed scrambler canbe accomplished
MSE is used to evaluate the amount of differencesbetween plain image and its corresponding scrambled one[48] MSE can be numerically evaluated using (4) with theexpected result being a value as high as possible thus denoting
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
(a)
R0
R1
R2
R3R4
R5
R6
R7
(b)
I0(i minus 2 j minus 1) I0(i minus 2 j + 1)
I0(i minus 1 j + 2)
I0(i + 1 j + 2)
I0(i + 2 j + 1)I0(i + 2 j minus 1)
I0(i + 1 j minus 2)
I0(i minus 1 j minus 2)
I0(i j)
(c)
Figure 1 Knightrsquos moving rules (a) Knightrsquos possible moves (b) movesrsquo possible annotations (ie as rules) and (c) rulesrsquo associatedcoordinates
chaotic map and description of the main procedures andalgorithms)
21 Use of Knightrsquos Moving Rules in Scramblerrsquos DesignKnightrsquos moving rules unlike those of all other pieces followan ldquoelbowedrdquo nonlinear shape (ie in formof letter 119871 in otherwords a horizontal displacement of two squares followed bya vertical displacement of one square or a vertical displace-ment of two squares followed by a horizontal displacement ofone square in either direction)Thus as shown in Figure 1(a)one can define a set of eight possiblemoves each of which canbe labeled as suggested in Figure 1(b)
Considering any 24-bit color bitmap 1198680 of the size 119898 times
119898 times 119896 where 119896 isin RGB represents imagersquos color channelas a chessboard with fixed orientation and any pixel 119868
0(119894 119895 119896)
within this image as knightrsquos initial position each movingrule (ie pixelrsquos new destination) can be mathematicallyexpressed in relative coordinates as shown in Figure 1(c)
Since a digital image scrambler aims among others tominimize adjacent pixelsrsquo correlation we have considered thefollowing slightly different approaches
(i) in order to ensure the fact that moved pixelsrsquo valueswill be less correlated with those of their new neigh-boring pixels any pixel 119868
0(119894 119895 119896) will not be moved
once but 119899 times using the same moving rule 119877119909
(ii) reduced susceptibility of correlation between movedpixelrsquos value and of new neighboring pixelsrsquo (ieachieved by applying previous approach) will bestrengthened by the fact that once pixelrsquos new coordi-nates (ie 119894 and 119895) have been determined accordingto a binary random variable 119875
119910119911 its value will be
simultaneously transposed between color channels
Obviously a series of further questions arises for exampleldquowhich are 119899rsquos limits of variationsrdquo and ldquowhat happens ifpixelrsquos new destinationrsquos coordinates exceed imagersquos limitsrdquoor ldquoin what way the random binary variable 119875
119910119911influences
pixelsrsquo transposition between imagersquos color channelsrdquoThe answers to these questions (without any restriction
ie one can think of other solutions) are as follows
(i) from a theoretical perspective 119899 can accommodateany positive integer value however in practical terms(related to PRBGrsquos use as it will be discussed furtherwithin this section) its value will be restricted to an8-bit maximum value (ie 0 le 119899 le 255)
(ii) indeed there is an imminent possibility that pixelsrsquonew destinationrsquos coordinates exceed imagersquos limits(especially when pixels to be moved are situated atimagersquos extreme limits or when 119899rsquos value is close tomaximum) and therefore considering the image asbeing virtually cyclic 119894 and 119895 coordinates are to bekept within [1 119898] interval (ie in a relatively plasticbut more proper expression ldquoexitingrdquo through oneside of the image the pixel will ldquoreturnrdquo within theimage on the other side completing ldquoin a natural wayrdquomotionrsquos specific shape)
(iii) when it comes to transpose pixelsrsquo values betweenimagersquos color channels the basic idea is to ensure thatall pixels are removed from their initial color plan(eg considering any pixel 119868
0(119894 119895R) ie belonging
to the red plan its destination color plane will beforced to green or blue ie 119896 isin GB) Thiscriterion is satisfied by applying for example (ienotwithstanding the fact that other rules may beproposed) transposing rules presented in Figure 2
22 Use of the Chaotic Map in Scramblerrsquos Design Althoughany PRBG can be used for example [34ndash40] due to its goodcryptographic properties and suitability for cryptographicapplication that is large key space and good randomnessfeatures as proven in [41] PRNG model (1) was usedwithin designing and testing stages of this work On PRNGrsquosoutput sequence of real numbers themultilevel discretizationmethod [42] (eg with four thresholds ie 2-bit encoding ofeach interval) was applied resulted dibits being spread intotwo different files (ie ldquoBits Atxtrdquo containing dibitrsquos first bitand resp ldquoBits Btxtrdquo containing dibitrsquos second bit)
Using random sequences of real numbers generated bythe orbits of 119891
119879(1) designed based on a binary composition
Mathematical Problems in Engineering 3
G BRGBR
GB RGBR
0 1
0
1
Pz
Py
(a)
G BR GBR
GBR GBR
0 1
0
1
Pz
Py
(b)
0 1
0
1
GBR
GBR
GBR
Pz
Py
GB R
(c)
Figure 2 Correlation between the random binary variable 119875119910119911
and pixelsrsquo transposing rules when working on pixelsrsquo from (a) red colorchannel (b) blue color channel and (c) green color channel
Require m k imagersquos dimensionsEnsure 119888 a temporary counter initially set to 0
KMR a matrix of the size119898 sdot 119898 sdot 119896 initially set to 0 (ie KMR = zeros (119898119898 119896))RNS a matrix of the size119898 sdot 119898 sdot 119896 initially set to 0 (ie RNS = zeros (119898119898 119896))
for 119901 =1 kfor 119902 =1 m
for 119903 = 1 n119861119910119905119890119860= 119904119905119903119888119886119905 (Bits Atxt (119888 + 119904)) 119904 = 1 8 takes 8 bits from file Bits A
119861119910119905119890119861= 119904119905119903119888119886119905 (Bits Btxt (119888 + 119904)) 119904 = 1 8 takes 8 bits from file Bits B
KMR (119894 119895 119896) = 1198871198941198992119889119890119888 (119861119910119905119890119860) update KMR matrix
RNS (119894 119895 119896) = 1198871198941198992119889119890119888 (119861119910119905119890119861) update RNS matrix
119888 = 119888 + 8 update the temporary counterend
endend
Procedure 1 Computing KMR and RNS (Bits A Bits Bm k)
(2) of two identical one-dimensional chaotic discrete dynam-ical systems of form (22) in conjunction with the method ofdiscretization previously referenced 119898 sdot 119898 sdot 119896 sdot 8 dibit pairshave been generated (ie 6291456 bits were written in eachfile) this number being as seen directly proportional to theimage dimensions Consider
119891119879= 1198911(1199091
119894 1199031) lowast 1198912(1199092
119894 1199032) =
1198911(1199091
119894 1199031) + 1198912(1199092
119894 1199032)
1 minus 1198911(1199091
119894 1199031) sdot 1198912(1199092
119894 1199032)
(1)
where 11990910 11990920are the initial conditions 119903
1 1199032are the control
parameters and 1199091
119894 1199092119894are the two orbits obtained by
recurrences 1199091119894+1
= 1198911(1199091
119894 1199031) and 1199092
119894+1= 1198912(1199092
119894 1199032) for any
119894 isin 0 1 2 Consider
119886 lowast 119887 =119886 + 119887
1 minus 119886 sdot 119887 (2)
1198911 [minus1 1] 997888rarr [minus1 1]
1198911(119909 1199031) =
2
120587arctg (ctg (119903
1sdot 119909)) 119903
1isin [1 10]
1198912 [minus1 1] 997888rarr [minus1 1]
1198912(119909 1199032) =
2
120587arctg (ctg (119903
2sdot 119909)) 119903
2isin [1 10]
(3)
In our scramblerrsquos design and during subsequent tests119891119879rsquos initial seeding points respectively control parametersrsquo
values were chosen as follows 11990910= 0687754925117 and
1199031= 5938725025421 respectively 1199092
0= minus0013462335467
and 1199032= 1237490188615
Under the previous circumstances two matrices werecomputed hereafter referred to as KMR and RNS (ie theone associated with knightrsquos moving rules and respectivelythe one associated with the number of steps to be appliedon each rule) Procedure 1 describes how KMR and RNS arecomputed
KMR and RNS matrices are used to establish values of119877119909 119875119910119911 and 119899 as Figure 3 suggests It can be noticed that
we are dealing with a two-step algorithm Thus accordingto 119875119910119911rsquos value during the first step pixel 119868
0(119894 119895 119896) is moved
119899119886times using knightrsquos rule 119877
119886and then transposed to other
color planes while during the second step with its new
4 Mathematical Problems in Engineering
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
X X X X X X X X
b0 b1 b2 b3 b4 b5 b6 b7
b0b1b2b3b4b5b6b7
nb
na
RNS(i j)
(b)
Figure 3 Method of determining 119877119909 119875119910119911 and 119899 values (a) extraction of 119877
119909and 119875
119910119911values and (b) establishing the number of steps for each
rule
Require 119894 119895 119896 current pixelrsquos (ie the one to be moved) coordinates119870Rule = 1198891198901198882119887119894119899(KMR(119894 119895 119896)119877119886= 119870Rule (1 3) 119877
119886= 1198871198941198992119889119890119888 (119877
119886) compute Knightrsquos rule 119877
119886
119877119887= 119870Rule (5 7) 119877
119887= 1198871198941198992119889119890119888 (119877
119887) compute Knightrsquos rule 119877
119887
119875119910= 119870Rule (4) 119875
119911= 119870Rule (8) compute 119875
119910119911binary variable used to establish transposition rule
119899119886= RNS(119894 119895 119896) number of steps to be applied using rule 119877
119886
119899119887= 1198891198901198882119887119894119899 (119899
119886)
119899119887= 119891119897119894119901119897119903 (119899
119887)
119899119887= 1198871198941198992119889119890119888 (119899
119887) number of steps to be applied using rule 119877
119887
Procedure 2 Extracting features(KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
coordinates the pixel is moved 119899119887times using knightrsquos rule
119877119887and then into the another color planeProcedure 2 shows how 119875
119910119911 119877119886 119899119886 119877119887 and 119899
119887are
computed for each imagersquos pixel
23 Practical Example In order to facilitate understandingof those previously described the example shown in Figure 4comes into aid of the reader For this example a colorbitmap of the size 12 times 12 times 3 a starting pixel 119868
0(3 6 1)
119875119910119911
= 0 0 1198773as first stage rule 119877
5as second stage
rule usage of 1198773three times and usage of 119877
5five times are
considered
24 Other Useful Procedures Before starting with thedescription of Algorithms 1 and 2 two more procedures mustbe properly described namely the one which deals withcomputation of pixelsrsquo new coordinates both for scramblingand descrambling algorithms
Thus with 1198680representing the pixelsrsquo values matrix of a
24-bit color bitmap of the size 119898 times 119898 and 119899 representingnumber of steps to be appliedwith rule119877
119909 during scrambling
process pixelsrsquo new coordinates can be computed usingProcedure 3
On the basis thatmoving rules are symmetric (eg effectsof rule 119877
3are reversed by applying rule 119877
7) during descram-
bling process pixelsrsquo new coordinates can be computed usingthe same procedure but after modifying one of the functioncallrsquos parameters as Procedure 4 suggests
25 Scrambling and Descrambling Algorithmsrsquo DescriptionDepending on random binary variable 119875
119910119911value pixelsrsquo
color values are transposed between RGB channels usingProcedure 5 (ie during imagersquos scrambling process) andrespectively Procedure 6 (ie during imagersquos descramblingprocess)
3 Analysis and Comparison Results
As a general requirement for any digital image scrambler theoutput image should be greatly different in comparison withits plain version (ie from statistical point of view) To quan-tify this requirement in addition to visual assessment fewstatistical estimators can be used the most commonly onesbeing presented in the following subsections and accord-ing to an already widely used conventional methodology[43 44]
31 Visual Analysis The purpose of visual testing is tohighlight presence of similarities between plain image and itsscrambled version (ie if the scrambled image does or doesnot contain any features of the plain image) For the proposedscrambler same as for all digital image scramblers expecta-tions are to rearrange plain imagersquos pixels in a deterministicway but with a random-like appearance
Visual testing was performed on the 512 times 512 pixels24-bit Lena Peppers and Baboon color bitmaps from theUSC-SIPI miscellaneous image dataset [45] Figure 5 depicts
Mathematical Problems in Engineering 5
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 = 1 k
for 119902 = 1 mfor 119903 = 1 m
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119886
0(119894 119895 119896) = Compute limits on scrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
119868119887
0(119894 119895 119896) = Compute limits on scrambling (119868119886
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
Transpose pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 1 Scrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 =k minus1 1
for 119902 =m minus1 1for 119903 =m minus1 1
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119887
0(119894 119895 119896) = Compute limits on descrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
119868119886
0(119894 119895 119896) = Compute limits on descrambling (119868119887
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
Transpose pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 2 Descrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
these test plain images whereas their scrambled versionsare showcased in Figure 6 By comparing them one can saythat there is no perceptual similarity (ie no visual infor-mation can be observed in the processed versions of plainimages)
32 Adjacent Pixelsrsquo Correlation Coefficientsrsquo Analysis It iswell known the fact that generally in plain images anyarbitrarily chosen pixels are strongly correlated with theiradjacent ones (either they are diagonally vertically or hor-izontally oriented) [5 17 31] With respect to this idea an
efficient digital image scrambling schememustminimize thiscorrelation as much as possible
Whilst Figure 7 showcases the correlation distributionsof horizontally (a) vertically (b) and diagonally (c)adjacent pixels for the Baboon plain image Figure 8showcases the correlation distributions (ie for thesame pixelsrsquo adjacency cases) for the Baboon scrambledimage
At the same time all APCCs (computed over 10000 pairsof adjacent pixels randomly selected for each of the testingdirections and for each of the color channels) are summarizedin Table 1 for each of the test images
6 Mathematical Problems in Engineering
1
2
3
(a)
1 2 3
4
5
(b)
(c)
Figure 4 Pixel scramblingtransposition example (a) First stagersquos output image (b) second stagersquos output image and (c) final output image
(a) (b) (c)
Figure 5 Test plain images used during algorithmrsquos testing procedures (a) Lena (b) Peppers and (c) Baboon
It can be easily noticed that neighboring pixels in theplain images are highly correlated that is APCCsrsquo valuesare too high very close to one On the contrary in cases ofscrambled images those values are close to zero meaningthat all neighboring pixels considered in tests are weaklycorrelated which is the expected result [5 46 47]
33 Other Qualitative Measurementsrsquo Analysis Whereasthrough the visual assessment and APCCsrsquo analysis good
scrambling effects are highlighted based on MSE (meansquared error) NPCR (number of pixel change rate) andUACI (unified average changing intensity) measures a betterandmore objective assessment of the proposed scrambler canbe accomplished
MSE is used to evaluate the amount of differencesbetween plain image and its corresponding scrambled one[48] MSE can be numerically evaluated using (4) with theexpected result being a value as high as possible thus denoting
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
G BRGBR
GB RGBR
0 1
0
1
Pz
Py
(a)
G BR GBR
GBR GBR
0 1
0
1
Pz
Py
(b)
0 1
0
1
GBR
GBR
GBR
Pz
Py
GB R
(c)
Figure 2 Correlation between the random binary variable 119875119910119911
and pixelsrsquo transposing rules when working on pixelsrsquo from (a) red colorchannel (b) blue color channel and (c) green color channel
Require m k imagersquos dimensionsEnsure 119888 a temporary counter initially set to 0
KMR a matrix of the size119898 sdot 119898 sdot 119896 initially set to 0 (ie KMR = zeros (119898119898 119896))RNS a matrix of the size119898 sdot 119898 sdot 119896 initially set to 0 (ie RNS = zeros (119898119898 119896))
for 119901 =1 kfor 119902 =1 m
for 119903 = 1 n119861119910119905119890119860= 119904119905119903119888119886119905 (Bits Atxt (119888 + 119904)) 119904 = 1 8 takes 8 bits from file Bits A
119861119910119905119890119861= 119904119905119903119888119886119905 (Bits Btxt (119888 + 119904)) 119904 = 1 8 takes 8 bits from file Bits B
KMR (119894 119895 119896) = 1198871198941198992119889119890119888 (119861119910119905119890119860) update KMR matrix
RNS (119894 119895 119896) = 1198871198941198992119889119890119888 (119861119910119905119890119861) update RNS matrix
119888 = 119888 + 8 update the temporary counterend
endend
Procedure 1 Computing KMR and RNS (Bits A Bits Bm k)
(2) of two identical one-dimensional chaotic discrete dynam-ical systems of form (22) in conjunction with the method ofdiscretization previously referenced 119898 sdot 119898 sdot 119896 sdot 8 dibit pairshave been generated (ie 6291456 bits were written in eachfile) this number being as seen directly proportional to theimage dimensions Consider
119891119879= 1198911(1199091
119894 1199031) lowast 1198912(1199092
119894 1199032) =
1198911(1199091
119894 1199031) + 1198912(1199092
119894 1199032)
1 minus 1198911(1199091
119894 1199031) sdot 1198912(1199092
119894 1199032)
(1)
where 11990910 11990920are the initial conditions 119903
1 1199032are the control
parameters and 1199091
119894 1199092119894are the two orbits obtained by
recurrences 1199091119894+1
= 1198911(1199091
119894 1199031) and 1199092
119894+1= 1198912(1199092
119894 1199032) for any
119894 isin 0 1 2 Consider
119886 lowast 119887 =119886 + 119887
1 minus 119886 sdot 119887 (2)
1198911 [minus1 1] 997888rarr [minus1 1]
1198911(119909 1199031) =
2
120587arctg (ctg (119903
1sdot 119909)) 119903
1isin [1 10]
1198912 [minus1 1] 997888rarr [minus1 1]
1198912(119909 1199032) =
2
120587arctg (ctg (119903
2sdot 119909)) 119903
2isin [1 10]
(3)
In our scramblerrsquos design and during subsequent tests119891119879rsquos initial seeding points respectively control parametersrsquo
values were chosen as follows 11990910= 0687754925117 and
1199031= 5938725025421 respectively 1199092
0= minus0013462335467
and 1199032= 1237490188615
Under the previous circumstances two matrices werecomputed hereafter referred to as KMR and RNS (ie theone associated with knightrsquos moving rules and respectivelythe one associated with the number of steps to be appliedon each rule) Procedure 1 describes how KMR and RNS arecomputed
KMR and RNS matrices are used to establish values of119877119909 119875119910119911 and 119899 as Figure 3 suggests It can be noticed that
we are dealing with a two-step algorithm Thus accordingto 119875119910119911rsquos value during the first step pixel 119868
0(119894 119895 119896) is moved
119899119886times using knightrsquos rule 119877
119886and then transposed to other
color planes while during the second step with its new
4 Mathematical Problems in Engineering
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
X X X X X X X X
b0 b1 b2 b3 b4 b5 b6 b7
b0b1b2b3b4b5b6b7
nb
na
RNS(i j)
(b)
Figure 3 Method of determining 119877119909 119875119910119911 and 119899 values (a) extraction of 119877
119909and 119875
119910119911values and (b) establishing the number of steps for each
rule
Require 119894 119895 119896 current pixelrsquos (ie the one to be moved) coordinates119870Rule = 1198891198901198882119887119894119899(KMR(119894 119895 119896)119877119886= 119870Rule (1 3) 119877
119886= 1198871198941198992119889119890119888 (119877
119886) compute Knightrsquos rule 119877
119886
119877119887= 119870Rule (5 7) 119877
119887= 1198871198941198992119889119890119888 (119877
119887) compute Knightrsquos rule 119877
119887
119875119910= 119870Rule (4) 119875
119911= 119870Rule (8) compute 119875
119910119911binary variable used to establish transposition rule
119899119886= RNS(119894 119895 119896) number of steps to be applied using rule 119877
119886
119899119887= 1198891198901198882119887119894119899 (119899
119886)
119899119887= 119891119897119894119901119897119903 (119899
119887)
119899119887= 1198871198941198992119889119890119888 (119899
119887) number of steps to be applied using rule 119877
119887
Procedure 2 Extracting features(KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
coordinates the pixel is moved 119899119887times using knightrsquos rule
119877119887and then into the another color planeProcedure 2 shows how 119875
119910119911 119877119886 119899119886 119877119887 and 119899
119887are
computed for each imagersquos pixel
23 Practical Example In order to facilitate understandingof those previously described the example shown in Figure 4comes into aid of the reader For this example a colorbitmap of the size 12 times 12 times 3 a starting pixel 119868
0(3 6 1)
119875119910119911
= 0 0 1198773as first stage rule 119877
5as second stage
rule usage of 1198773three times and usage of 119877
5five times are
considered
24 Other Useful Procedures Before starting with thedescription of Algorithms 1 and 2 two more procedures mustbe properly described namely the one which deals withcomputation of pixelsrsquo new coordinates both for scramblingand descrambling algorithms
Thus with 1198680representing the pixelsrsquo values matrix of a
24-bit color bitmap of the size 119898 times 119898 and 119899 representingnumber of steps to be appliedwith rule119877
119909 during scrambling
process pixelsrsquo new coordinates can be computed usingProcedure 3
On the basis thatmoving rules are symmetric (eg effectsof rule 119877
3are reversed by applying rule 119877
7) during descram-
bling process pixelsrsquo new coordinates can be computed usingthe same procedure but after modifying one of the functioncallrsquos parameters as Procedure 4 suggests
25 Scrambling and Descrambling Algorithmsrsquo DescriptionDepending on random binary variable 119875
119910119911value pixelsrsquo
color values are transposed between RGB channels usingProcedure 5 (ie during imagersquos scrambling process) andrespectively Procedure 6 (ie during imagersquos descramblingprocess)
3 Analysis and Comparison Results
As a general requirement for any digital image scrambler theoutput image should be greatly different in comparison withits plain version (ie from statistical point of view) To quan-tify this requirement in addition to visual assessment fewstatistical estimators can be used the most commonly onesbeing presented in the following subsections and accord-ing to an already widely used conventional methodology[43 44]
31 Visual Analysis The purpose of visual testing is tohighlight presence of similarities between plain image and itsscrambled version (ie if the scrambled image does or doesnot contain any features of the plain image) For the proposedscrambler same as for all digital image scramblers expecta-tions are to rearrange plain imagersquos pixels in a deterministicway but with a random-like appearance
Visual testing was performed on the 512 times 512 pixels24-bit Lena Peppers and Baboon color bitmaps from theUSC-SIPI miscellaneous image dataset [45] Figure 5 depicts
Mathematical Problems in Engineering 5
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 = 1 k
for 119902 = 1 mfor 119903 = 1 m
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119886
0(119894 119895 119896) = Compute limits on scrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
119868119887
0(119894 119895 119896) = Compute limits on scrambling (119868119886
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
Transpose pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 1 Scrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 =k minus1 1
for 119902 =m minus1 1for 119903 =m minus1 1
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119887
0(119894 119895 119896) = Compute limits on descrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
119868119886
0(119894 119895 119896) = Compute limits on descrambling (119868119887
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
Transpose pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 2 Descrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
these test plain images whereas their scrambled versionsare showcased in Figure 6 By comparing them one can saythat there is no perceptual similarity (ie no visual infor-mation can be observed in the processed versions of plainimages)
32 Adjacent Pixelsrsquo Correlation Coefficientsrsquo Analysis It iswell known the fact that generally in plain images anyarbitrarily chosen pixels are strongly correlated with theiradjacent ones (either they are diagonally vertically or hor-izontally oriented) [5 17 31] With respect to this idea an
efficient digital image scrambling schememustminimize thiscorrelation as much as possible
Whilst Figure 7 showcases the correlation distributionsof horizontally (a) vertically (b) and diagonally (c)adjacent pixels for the Baboon plain image Figure 8showcases the correlation distributions (ie for thesame pixelsrsquo adjacency cases) for the Baboon scrambledimage
At the same time all APCCs (computed over 10000 pairsof adjacent pixels randomly selected for each of the testingdirections and for each of the color channels) are summarizedin Table 1 for each of the test images
6 Mathematical Problems in Engineering
1
2
3
(a)
1 2 3
4
5
(b)
(c)
Figure 4 Pixel scramblingtransposition example (a) First stagersquos output image (b) second stagersquos output image and (c) final output image
(a) (b) (c)
Figure 5 Test plain images used during algorithmrsquos testing procedures (a) Lena (b) Peppers and (c) Baboon
It can be easily noticed that neighboring pixels in theplain images are highly correlated that is APCCsrsquo valuesare too high very close to one On the contrary in cases ofscrambled images those values are close to zero meaningthat all neighboring pixels considered in tests are weaklycorrelated which is the expected result [5 46 47]
33 Other Qualitative Measurementsrsquo Analysis Whereasthrough the visual assessment and APCCsrsquo analysis good
scrambling effects are highlighted based on MSE (meansquared error) NPCR (number of pixel change rate) andUACI (unified average changing intensity) measures a betterandmore objective assessment of the proposed scrambler canbe accomplished
MSE is used to evaluate the amount of differencesbetween plain image and its corresponding scrambled one[48] MSE can be numerically evaluated using (4) with theexpected result being a value as high as possible thus denoting
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
X X X X X X X X
b0 b1 b2 b3 b4 b5 b6 b7
b0b1b2b3b4b5b6b7
nb
na
RNS(i j)
(b)
Figure 3 Method of determining 119877119909 119875119910119911 and 119899 values (a) extraction of 119877
119909and 119875
119910119911values and (b) establishing the number of steps for each
rule
Require 119894 119895 119896 current pixelrsquos (ie the one to be moved) coordinates119870Rule = 1198891198901198882119887119894119899(KMR(119894 119895 119896)119877119886= 119870Rule (1 3) 119877
119886= 1198871198941198992119889119890119888 (119877
119886) compute Knightrsquos rule 119877
119886
119877119887= 119870Rule (5 7) 119877
119887= 1198871198941198992119889119890119888 (119877
119887) compute Knightrsquos rule 119877
119887
119875119910= 119870Rule (4) 119875
119911= 119870Rule (8) compute 119875
119910119911binary variable used to establish transposition rule
119899119886= RNS(119894 119895 119896) number of steps to be applied using rule 119877
119886
119899119887= 1198891198901198882119887119894119899 (119899
119886)
119899119887= 119891119897119894119901119897119903 (119899
119887)
119899119887= 1198871198941198992119889119890119888 (119899
119887) number of steps to be applied using rule 119877
119887
Procedure 2 Extracting features(KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
coordinates the pixel is moved 119899119887times using knightrsquos rule
119877119887and then into the another color planeProcedure 2 shows how 119875
119910119911 119877119886 119899119886 119877119887 and 119899
119887are
computed for each imagersquos pixel
23 Practical Example In order to facilitate understandingof those previously described the example shown in Figure 4comes into aid of the reader For this example a colorbitmap of the size 12 times 12 times 3 a starting pixel 119868
0(3 6 1)
119875119910119911
= 0 0 1198773as first stage rule 119877
5as second stage
rule usage of 1198773three times and usage of 119877
5five times are
considered
24 Other Useful Procedures Before starting with thedescription of Algorithms 1 and 2 two more procedures mustbe properly described namely the one which deals withcomputation of pixelsrsquo new coordinates both for scramblingand descrambling algorithms
Thus with 1198680representing the pixelsrsquo values matrix of a
24-bit color bitmap of the size 119898 times 119898 and 119899 representingnumber of steps to be appliedwith rule119877
119909 during scrambling
process pixelsrsquo new coordinates can be computed usingProcedure 3
On the basis thatmoving rules are symmetric (eg effectsof rule 119877
3are reversed by applying rule 119877
7) during descram-
bling process pixelsrsquo new coordinates can be computed usingthe same procedure but after modifying one of the functioncallrsquos parameters as Procedure 4 suggests
25 Scrambling and Descrambling Algorithmsrsquo DescriptionDepending on random binary variable 119875
119910119911value pixelsrsquo
color values are transposed between RGB channels usingProcedure 5 (ie during imagersquos scrambling process) andrespectively Procedure 6 (ie during imagersquos descramblingprocess)
3 Analysis and Comparison Results
As a general requirement for any digital image scrambler theoutput image should be greatly different in comparison withits plain version (ie from statistical point of view) To quan-tify this requirement in addition to visual assessment fewstatistical estimators can be used the most commonly onesbeing presented in the following subsections and accord-ing to an already widely used conventional methodology[43 44]
31 Visual Analysis The purpose of visual testing is tohighlight presence of similarities between plain image and itsscrambled version (ie if the scrambled image does or doesnot contain any features of the plain image) For the proposedscrambler same as for all digital image scramblers expecta-tions are to rearrange plain imagersquos pixels in a deterministicway but with a random-like appearance
Visual testing was performed on the 512 times 512 pixels24-bit Lena Peppers and Baboon color bitmaps from theUSC-SIPI miscellaneous image dataset [45] Figure 5 depicts
Mathematical Problems in Engineering 5
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 = 1 k
for 119902 = 1 mfor 119903 = 1 m
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119886
0(119894 119895 119896) = Compute limits on scrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
119868119887
0(119894 119895 119896) = Compute limits on scrambling (119868119886
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
Transpose pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 1 Scrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 =k minus1 1
for 119902 =m minus1 1for 119903 =m minus1 1
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119887
0(119894 119895 119896) = Compute limits on descrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
119868119886
0(119894 119895 119896) = Compute limits on descrambling (119868119887
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
Transpose pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 2 Descrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
these test plain images whereas their scrambled versionsare showcased in Figure 6 By comparing them one can saythat there is no perceptual similarity (ie no visual infor-mation can be observed in the processed versions of plainimages)
32 Adjacent Pixelsrsquo Correlation Coefficientsrsquo Analysis It iswell known the fact that generally in plain images anyarbitrarily chosen pixels are strongly correlated with theiradjacent ones (either they are diagonally vertically or hor-izontally oriented) [5 17 31] With respect to this idea an
efficient digital image scrambling schememustminimize thiscorrelation as much as possible
Whilst Figure 7 showcases the correlation distributionsof horizontally (a) vertically (b) and diagonally (c)adjacent pixels for the Baboon plain image Figure 8showcases the correlation distributions (ie for thesame pixelsrsquo adjacency cases) for the Baboon scrambledimage
At the same time all APCCs (computed over 10000 pairsof adjacent pixels randomly selected for each of the testingdirections and for each of the color channels) are summarizedin Table 1 for each of the test images
6 Mathematical Problems in Engineering
1
2
3
(a)
1 2 3
4
5
(b)
(c)
Figure 4 Pixel scramblingtransposition example (a) First stagersquos output image (b) second stagersquos output image and (c) final output image
(a) (b) (c)
Figure 5 Test plain images used during algorithmrsquos testing procedures (a) Lena (b) Peppers and (c) Baboon
It can be easily noticed that neighboring pixels in theplain images are highly correlated that is APCCsrsquo valuesare too high very close to one On the contrary in cases ofscrambled images those values are close to zero meaningthat all neighboring pixels considered in tests are weaklycorrelated which is the expected result [5 46 47]
33 Other Qualitative Measurementsrsquo Analysis Whereasthrough the visual assessment and APCCsrsquo analysis good
scrambling effects are highlighted based on MSE (meansquared error) NPCR (number of pixel change rate) andUACI (unified average changing intensity) measures a betterandmore objective assessment of the proposed scrambler canbe accomplished
MSE is used to evaluate the amount of differencesbetween plain image and its corresponding scrambled one[48] MSE can be numerically evaluated using (4) with theexpected result being a value as high as possible thus denoting
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Mathematical Problems in Engineering 5
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 = 1 k
for 119902 = 1 mfor 119903 = 1 m
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119886
0(119894 119895 119896) = Compute limits on scrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
119868119887
0(119894 119895 119896) = Compute limits on scrambling (119868119886
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
Transpose pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 1 Scrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
Require Bits Atxt and Bits Btxt files containing bitstreams generated by 119891119879rsquos trajectories
Compute KMR and RNS (Bits A Bits B m k) compute KMR and RNS matricesfor 119901 =k minus1 1
for 119902 =m minus1 1for 119903 =m minus1 1
Extract features (KMR (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
RNS (119894 119895 119896) 1le119894le1198981le119895le119898
1le119896le3
)
119868119887
0(119894 119895 119896) = Compute limits on descrambling (119868
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119887 119899119887)
119868119886
0(119894 119895 119896) = Compute limits on descrambling (119868119887
0(119894 119895 119896) 1le119894lt119898
1le119895lt119898
119877119886 119899119886)
Transpose pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
endend
end
Algorithm 2 Descrambling image(1198680(119894 119895 119896) 1le119894le119898
1le119895le119898
1le119896le3
)
these test plain images whereas their scrambled versionsare showcased in Figure 6 By comparing them one can saythat there is no perceptual similarity (ie no visual infor-mation can be observed in the processed versions of plainimages)
32 Adjacent Pixelsrsquo Correlation Coefficientsrsquo Analysis It iswell known the fact that generally in plain images anyarbitrarily chosen pixels are strongly correlated with theiradjacent ones (either they are diagonally vertically or hor-izontally oriented) [5 17 31] With respect to this idea an
efficient digital image scrambling schememustminimize thiscorrelation as much as possible
Whilst Figure 7 showcases the correlation distributionsof horizontally (a) vertically (b) and diagonally (c)adjacent pixels for the Baboon plain image Figure 8showcases the correlation distributions (ie for thesame pixelsrsquo adjacency cases) for the Baboon scrambledimage
At the same time all APCCs (computed over 10000 pairsof adjacent pixels randomly selected for each of the testingdirections and for each of the color channels) are summarizedin Table 1 for each of the test images
6 Mathematical Problems in Engineering
1
2
3
(a)
1 2 3
4
5
(b)
(c)
Figure 4 Pixel scramblingtransposition example (a) First stagersquos output image (b) second stagersquos output image and (c) final output image
(a) (b) (c)
Figure 5 Test plain images used during algorithmrsquos testing procedures (a) Lena (b) Peppers and (c) Baboon
It can be easily noticed that neighboring pixels in theplain images are highly correlated that is APCCsrsquo valuesare too high very close to one On the contrary in cases ofscrambled images those values are close to zero meaningthat all neighboring pixels considered in tests are weaklycorrelated which is the expected result [5 46 47]
33 Other Qualitative Measurementsrsquo Analysis Whereasthrough the visual assessment and APCCsrsquo analysis good
scrambling effects are highlighted based on MSE (meansquared error) NPCR (number of pixel change rate) andUACI (unified average changing intensity) measures a betterandmore objective assessment of the proposed scrambler canbe accomplished
MSE is used to evaluate the amount of differencesbetween plain image and its corresponding scrambled one[48] MSE can be numerically evaluated using (4) with theexpected result being a value as high as possible thus denoting
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
1
2
3
(a)
1 2 3
4
5
(b)
(c)
Figure 4 Pixel scramblingtransposition example (a) First stagersquos output image (b) second stagersquos output image and (c) final output image
(a) (b) (c)
Figure 5 Test plain images used during algorithmrsquos testing procedures (a) Lena (b) Peppers and (c) Baboon
It can be easily noticed that neighboring pixels in theplain images are highly correlated that is APCCsrsquo valuesare too high very close to one On the contrary in cases ofscrambled images those values are close to zero meaningthat all neighboring pixels considered in tests are weaklycorrelated which is the expected result [5 46 47]
33 Other Qualitative Measurementsrsquo Analysis Whereasthrough the visual assessment and APCCsrsquo analysis good
scrambling effects are highlighted based on MSE (meansquared error) NPCR (number of pixel change rate) andUACI (unified average changing intensity) measures a betterandmore objective assessment of the proposed scrambler canbe accomplished
MSE is used to evaluate the amount of differencesbetween plain image and its corresponding scrambled one[48] MSE can be numerically evaluated using (4) with theexpected result being a value as high as possible thus denoting
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Require 119894 119895 are current pixelrsquos coordinates regardless of the color channel in which it liesRequire 119909 represents the moving rule to be applied where 119909 isin 0 1 2 3 4 5 6 7Require 119899 represents the number of steps (ie how many times the rule is applied)switch 119909
case 0119894 = 119894 minus 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 1119894 = 119894 minus 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 2119894 = 119894 + 1 sdot 119899 119895 = 119895 + 2 sdot 119899
case 3119894 = 119894 + 2 sdot 119899 119895 = 119895 + 1 sdot 119899
case 4119894 = 119894 + 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
case 5119894 = 119894 + 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
case 6119894 = 119894 minus 1 sdot 119899 119895 = 119895 minus 2 sdot 119899
otherwise119894 = 119894 minus 2 sdot 119899 119895 = 119895 minus 1 sdot 119899
endRemark 119894 119895 represent pixelrsquos new coordinatesEnsure forall119894 forall119895 isin [1119898]
switch 119894
case 119894 lt 0119894 = 119894 + 512
otherwise119894 = mod (119894 119898) + 1 ensures 119894 gt 0
endswitch 119895
case 119895 lt 0119895 = 119895 + 512
otherwise119895 = mod (119895 119898) + 1 ensures 119895 gt 0
end
Procedure 3 Computing limits on scrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
Require 119909 represents the moving rule applied on scrambling procedure119909 = (mod (119909 + 4) 8)Remark 119909 represents the symmetric moving rule which is to be applied on descrambling procedure
Compute limits on scrambling (1198680(119894 119895) 1le119894lt1198981le119895lt119898
119877119909 119899)
Procedure 4 Computing limits on descrambling (1198680(119894 119895 ) 1le119894lt119898
1le119895lt119898
119877119909 119899)
nonidentical images (ie the higher the value of the MSEis the greater the differences between the two images are)Consider
MSE119862(119868119875 119868119878 119862) =
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
[119868119875(119894 119895 119862) minus 119868
119878(119894 119895 119862)]
2
(4)
where 119868119875
represents plain imagersquos associated matrix 119868119878
represents scrambled imagersquos associated matrix 119882 and 119867
represent the image dimensions (ie width and height) and119862 is the color channel (ie 119862 isin RGB equiv 1 2 3)
First shown in [49] and [50] and afterwards extensivelystudied [51] and presented in transposed fashion (moresuitable for usage within scrambled imagesrsquo assessment) [17]NPCR andUACImeasures are designed to estimate themean
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 5 Transposing pixel color values on scrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
(a) (b) (c)
Figure 6 Image scrambling results (a) Scrambled version of Lena (b) scrambled version of Peppers and (c) scrambled version of Baboon
number of distinct pixels having the same position in theplain image as in corresponding scrambled one respectivelyto estimate the average intensity differences of distinct pixelshaving the same position in the plain image as in thecorresponding scrambledone
Defined for each of imagersquos color channels [49ndash51]NPCR indicator can be numerically evaluated using (5)and considering two random images (ie the plain image
is completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[NPCR
119862] = 996093 Consider
NPCR119862(119868119875 119868119878 119862) = (
1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
119863(119894 119895 119862)) times 100
(5)
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Require 119875119910 119875119911
the random binary value which establish the transposing rule119896 current working color channel
Ensure Pixel Temporary variablePixel = 119868
0(119894 119895 119896)
switch (119875119910 119875119911 119896)
case (0 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 2) 119868
119887
0(119894 119895 119896 + 2) = Pixel
case (1 0 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 + 2) 119868
119886
0(119894 119895 119896 + 2) = Pixel
case (1 1 1)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = Pixel
case (0 0 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
case (0 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 + 1) 119868
119887
0(119894 119895 119896 + 1) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (1 0 2)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (1 1 2)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 + 1) 119868
119886
0(119894 119895 119896 + 1) = Pixel
case (0 0 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 2) 119868
119887
0(119894 119895 119896 minus 2) = 119868
119886
0(119894 119895 119896 minus 1) 119868
119886
0(119894 119895 119896 minus 1) = Pixel
case (0 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 0 3)1198680(119894 119895 119896) = 119868
119886
0(119894 119895 119896 minus 2) 119868
119886
0(119894 119895 119896 minus 2) = Pixel
case (1 1 3)1198680(119894 119895 119896) = 119868
119887
0(119894 119895 119896 minus 1) 119868
119887
0(119894 119895 119896 minus 1) = Pixel
end
Procedure 6 Transposing pixel color values on descrambling (1198680(119894 119895 119896) 119868
119886
0(119894 119895 119896) 119868
119887
0(119894 119895 119896) 119875
119910 119875119911 119896)
Table 1 Adjacent pixelsrsquo correlation coefficientsrsquo analysis
Pixelsrsquo adjacency Image status Color channel Test imagesLena Peppers Baboon
Horizontally
OriginalR 09785 09677 08722G 09747 09815 07743B 09695 09661 08857
ScrambledR 00063 00009 00011G 00110 00044 minus00009B 00104 00012 00015
Vertically
OriginalR 09899 09695 09241G 09877 09831 08680B 09842 09696 09088
ScrambledR 00004 minus00017 00002G minus00064 minus00022 00032B 00003 minus00020 00004
Diagonally
OriginalR 09685 09565 08624G 09622 09719 07441B 09533 09527 08420
ScrambledR minus00020 00071 minus00013G 00166 00007 00016B 00049 00035 minus00007
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
(a) (b) (c)
Figure 7 Correlation distributions of adjacent pixels in Baboon plain image (a) Horizontally adjacent pixels (b) vertically adjacent pixelsand (c) diagonally adjacent pixels
(a) (b) (c)
Figure 8 Correlation distribution of adjacent pixels in Baboon scrambled image (a) Horizontally adjacent pixels (b) vertically adjacentpixels and (c) diagonally adjacent pixels
where 119863(119894 119895 119862) represents the matrix associated with thedifferences between 119868
119875rsquos and 119868
119878rsquos color channels with its struc-
ture being computed according to the following relationship
119863(119894 119895 119862) = 0 if 119868
119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
1 if 119868119875(119894 119895 119862) = 119868
119878(119894 119895 119862)
(6)
Defined for each of imagersquos color channels [49ndash51]UACI indicator can be numerically evaluated using (7)and considering two random images (ie the plain imageis completely different in comparison with its scrambledversion) its maximum expected value [17] is found to be120576[UACI
119862] = 334635 Consider
UACI119862(119868119875 119868119878 119862)
= (1
119882 times119867
119882
sum
119894=1
119867
sum
119895=1
1003816100381610038161003816 119868119875 (119894 119895 119862) minus 119868119878 (119894 119895 119862)1003816100381610038161003816
255) times 100
(7)
Analyzing NPCR UACI and MSE indicatorsrsquo values foreach of the test images as they are summarized inTable 2 onecan say that the proposed scrambling scheme approaches theperformance of an ideal digital image scrambler Howeverthe attention is drawn to UACI indicatorrsquos values not very
close to the maximum expected value Reasoning for this isthe fact that the proposed digital image scrambler keeps thesame set of possible values of pixels in the scrambled imageas in the plain image Obviously this limits the maximumdifference values that could be achieved between plain imagersquospixels and scrambled imagersquos corresponding ones [17] Forexample in case of the Peppers plain image maximumdifferences that could be achieved on each color channel areupper bounded by pixels maximum values (ie 231 for Rchannel 239 for G channel and 229 for B channel) insteadof the theoretical one of 255
34 Key Space and Key Sensitivity Analysis A good digitalimage scrambler should have a sufficiently large key spaceto resist brute-force attacks mostly if its usage as part ofimagesrsquo encryption schemes is desired (ie as part of the twofundamental techniques used to encrypt a block of pixelsie permutation stage which aims to ensure the covetedconfusion properties)
As previously shown the proposed digital image scram-bler operates based on values of the bitstreams generatedby a multithresholded digital chaotic map Thus taking intoconsideration the two seeding points respectively the twocontrol parameters of the digital chaotic generator (namely
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 2 Difference measures between original and scrambled images
Image Color channel MeasuresNPCR () UACI () MSE
LenaR 996059 329915 37081 sdot 103
G 996013 307161 30761 sdot 103
B 995735 302703 30090 sdot 103
PeppersR 995174 285412 26343 sdot 103
G 993126 319327 33309 sdot 103
B 992966 289468 27533 sdot 103
BaboonR 995007 252236 21084 sdot 103
G 994621 245648 18140 sdot 103
B 995296 268748 23806 sdot 103
R0
R1
R2
R3
R4
R5
R6
R7
(a)
R0 R1
R2 R3
R4 R5
R6 R7
(b)
R0
R1
R2
R3
R4
R5
R6
R7
(c)
Figure 9 Three more cases out of 8 of possible knightrsquos moving rules annotations
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(a)
PzPy
X X X X X X X X
Ra2 Ra0Ra1 Rb2 Rb0Rb1
KMR(i j)
(b)
Figure 10 Two more cases out of 8 of possible feature extraction vectorrsquos elements configuration
1199091
0 1199092
0 1199031 1199032) with a precision of up to 10
minus15 and the keyspace of a possible encryption scheme (which uses proposeddigital image scrambler scheme within permutation stage)is significantly improved by the key space of the proposedscrambler namely by a factor of 1060 (hereafter referred toas improvement factor and abbreviated as IF)
On its turn this key space improvement factor can beraised if some artifacts are considered such as (but notlimited to) the following
(i) if knightrsquos moving rules are to be permuted on userchoice (eg any of the three more cases out of 8listed in Figure 9 may be used) then IF increases with1046055
(ii) if features extraction vectorrsquos elements are to bepermuted on user choice (eg any of the two morecases out of 4 (ie if bits defining each rule are takengrouped) or out of 8 (ie if bits defining each rule aretaken random) listed in Figure 10) and may be usedthen IF increases with another 1013802 or 1046055
(iii) if color channels interchanging rules are to be perm-uted on user choice considering that there are(1198624
3)3
= 3375 possibleways to do it then IF increaseswith 1035283
Thereby proposed scramblerrsquos key space can achieve avalue of 10695140 (for the first case mentioned in (ii)) or
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
(a) (b) (c)
Figure 11 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using119870119878 (b) scrambled image using119870
119863 and (c) image
representing differences between (a) and (b)
(a) (b) (c)
Figure 12 Key sensitivity of the proposed digital image scrambler (a) Scrambled image using 119870119878 (b) descrambled image using 119870
119878 and
(c) descrambled image using 119870119863
of 10727393 (for the second case mentioned in (ii)) whichis sufficiently large to recommend usage of the proposedscrambler within any image encryption scheme
In addition in order to approach the performance of anideal image encryption algorithm the proposed digital imagescrambler should have high sensitivity to scrambling key (iea slight key change should lead to significant changes inthe scrambled images during scrambling procedure respec-tively significant changes in the descrambled images duringdescrambling procedure)
Figure 11 shows the key sensitivity of the proposed digitalimage scrambler when operating over Baboon plain imagewith two different scrambling keys (differentiated from eachother by the LSB of initial seeding points ie by plusmn1 sdot
10minus15 variations of 1199091
0and 119909
2
0) As can be seen one bit
change in the scrambling key leads to two different scrambledimages whose difference is also random-like When image isdescrambled using an incorrect key (eg with same variationconditions of 1199091
0and 119909
2
0) the result is also a random-like
image as shown in Figure 12
The two keys used for key sensitivity analysis of theproposed digital image scrambler are
119870119878= 1199091
0= 0687754925117 119903
1= 593872502542
1199092
0= minus0013462335467 119903
1= 1237490188615
119870119863= 1199091
0= 0687754925118 119903
1= 593872502542
1199092
0= minus0013462335466 119903
1= 1237490188615
(8)
35 Performancesrsquo Comparison with Other Digital ImageScrambling Schemes Performancesrsquo comparison was donetaking into consideration some of the most recent works [1719 27 52] Table 3 summarizes NPCRrsquos UACIrsquos APCCsrsquo andspeedsrsquo measure mean values computed over the 512 times 512pixels 24-bit and color bitmaps for all subjected scramblingalgorithms
Whilst the proposed digital image scrambler presentssimilar or better results in comparison with other proposed
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
Table 3 Mean values of NPCR UACI APCCs and speed measures
Measure Digital image scrambling algorithmDascalescu and Boriga [17] Dalhoum et al [19] Wu et al [27] Ye [52] Ours
NPCR 99431 90126 94163 93676 99489UACI 25032 NaN NaN NaN 29006APCC
Horizontal 00398 00767 00020 01059 00049Vertical 00448 01007 00022 01094 00018
Speed (KBs) 125 sdot 103 NaN 193502 NaN 117028
ones when it comes to general qualitative measures (ieNPCR UACI and APCCs) it seems to lack speed (ie fromprocessing time point of view) a fact which can be upheld toMATLABrsquos very slow for-loop execution But using parallelcomputations or (and) other programming languages theprocessing speed can be largely enhanced
All the above results (ie numerical results presentedin the entire Section 3) were generated with the proposedscrambling scripts for (de)shufflingwritten onMATLAB 730and run on an Intel Pentium Dual CPU T3200 at 200GHzpersonal computer
4 Conclusions
In the present paper the study of a newly designed digitalimage scrambler (as part of the two fundamental techniquesused to encrypt a block of pixels ie the permutationstage) that uses knightrsquos moving rules (ie from the game ofchess) in conjunction with a chaos-based PRBG in order totranspose original imagersquos pixels between RGB channels waspresented
Theoretical and practical arguments rounded by verygood numerical results on scramblerrsquos performance analysis(ie under various investigation methods including visualinspection adjacent pixelsrsquo correlation coefficientsrsquo com-putation keyrsquos space sensitivity assessment etc) confirmviability of the proposed scrambling method (ie it ensuresthe coveted confusion factor) recommending its usage withincryptographic applications thus contributing to the crystal-lization of reminded newparadigm (ie in the field of imagesrsquoencryption algorithms designing branch namely designingof new digital image scramblers based on rule sets of themostpopular games)
As futurework actual implementation onFPGA (focusedon the optimization of proposed algorithm for parallelcomputing) is concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] AMusheer andFOmar ldquoAmulti-level blocks scrambling basedchaotic image cipherrdquo in Proceedings of the 3rd International
Conference on Contemporary Computing pp 171ndash182 NoidaIndia August 2010
[2] S Rakesh A A Kaller B C Shadakshari et al ldquoImageencryption using block based uniform scrambling and chaoticlogistic mappingrdquo International Journal on Cryptography andInformation Security vol 2 no 1 pp 49ndash57 2012
[3] Q Zhang X Xue and X Wei ldquoA novel image encryptionalgorithm based onDNA subsequence operationrdquoThe ScientificWorld Journal vol 2012 Article ID 286741 10 pages 2012
[4] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009
[5] S Etemadi Borujeni andM Eshghi ldquoChaotic image encryptiondesign using tompkins-paige algorithmrdquo Mathematical Prob-lems in Engineering vol 2009 Article ID 762652 22 pages 2009
[6] H-F Huang ldquoPerceptual image watermarking algorithm basedonmagic squares scrambling in DWTrdquo in Proceedings of the 5thInternational Joint Conference on International Conference onNetworked Computing International Conference on AdvancedInformation Management and Service and International Con-ference on Digital Content Multimedia Technology and ItsApplications pp 1819ndash1822 Seoul Republic of Korea August2009
[7] R Ye ldquoA novel image scrambling and watermarking schemebased on orbits of Arnold Transformrdquo in Proceedings of thePacific-Asia Conference on Circuits Communications and Sys-tems pp 485ndash488 Chengdu China May 2009
[8] C R Wei J J Lid and G Y Liang ldquoA DCT-SVD domainwatermarking algorithm for digital image based on Moore-model cellular automata scramblingrdquo in Proceedings of theIEEE International Conference on Intelligent Computing andIntegrated Systems (ICISS rsquo10) pp 104ndash108 Guilin ChinaOctober 2010
[9] L Fang and W YuKai ldquoRestoring of the watermarking imagein Arnold scramblingrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems vol 1 pp 771ndash774Dalian China July 2010
[10] Y Li and X Hao ldquoA blind watermarking algorithm based onimage scrambling and error correct coding preprocessingrdquo inProceedings of the International Conference on Electrical andControl Engineering pp 4231ndash4233 Yichang China September2011
[11] G Liu H Liu and A Kadir ldquoWavelet-based color pathologicalimage watermark through dynamically adjusting the embed-ding intensityrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 406349 10 pages 2012
[12] V Seenivasagam and R Velumani ldquoA QR code based zero-watermarking scheme for authentication of medical images in
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
teleradiology cloudrdquoComputational andMathematicalMethodsin Medicine vol 2013 Article ID 516465 16 pages 2013
[13] S Li Y Zhao and B Qu ldquoImage scrambling based on chaoticsequences and Vigenere cipherrdquoMultimedia Tools and Applica-tions vol 66 no 3 pp 573ndash588 2013
[14] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
[15] C Guang-Hui H Kai and T Wei ldquoImage scrambling based onLogistic uniform distributionrdquo Acta Physica Sinica vol 60 no11 Article ID 110508 7 pages 2011
[16] G Zhang and Q Liu ldquoA novel image encryption method basedon total shuffling schemerdquoOptics Communications vol 284 no12 pp 2775ndash2780 2011
[17] A C Dascalescu and R E Boriga ldquoA novel fast chaos-basedalgorithm for generating random permutations with high shiftfactor suitable for image scramblingrdquo Nonlinear Dynamics vol74 no 1-2 pp 307ndash318 2013
[18] R Ye and H Li ldquoA novel image scrambling and watermarkingscheme based on cellular automatardquo in Proceedings of theInternational Symposium on Electronic Commerce and Securitypp 938ndash941 Guangzhou China August 2008
[19] A L A Dalhoum B A Mahafzah A A Awwad I AldamariA Ortega and M Alfonseca ldquoDigital image scrambling using2D cellular automatardquo IEEE Transactions onMultimedia vol 19no 4 pp 28ndash36 2012
[20] R Mathews A Goel P Saxena and V P Mishra ldquoImageencryption based on explosive inter-pixel displacement of theRGB attributes of a pixelrdquo in Proceedings of the World Congresson Engineering and Computer Science vol 1 pp 41ndash44 SanFrancisco Calif USA October 2011
[21] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel colorimage scrambling technique based on transposition of image-blocks between RGB color componentsrdquo International Journalof Research in Engineering amp Advanced Technology vol 1 no 2pp 1ndash8 2013
[22] P Jagadeesh P Nagabhushan and R P Kumar ldquoA novel imagescrambling technique based on information entropy and quadtree decompositionrdquo International Journal of Computer ScienceIssues vol 10 no 2 pp 285ndash294 2013
[23] G Ye X Huang and C Zhu ldquoImage encryption algorithm ofdouble scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007
[24] K T Lin ldquoHybrid encoding method by assembling the magic-matrix scrambling method and the binary encoding method inimage hidingrdquoOptics Communications vol 284 no 7 pp 1778ndash1784 2011
[25] D van de Ville W Philips R van de Walle and I LemahieuldquoImage scrambling without bandwidth expansionrdquo IEEE Trans-actions on Circuits and Systems for Video Technology vol 14 no6 pp 892ndash897 2004
[26] Y Zou X Tian S Xia and Y Song ldquoA novel image scramblingalgorithm based on Sudoku puzzlerdquo in Proceedings of the 4thInternational Congress on Image and Signal Processing (CISP rsquo11)vol 2 pp 737ndash740 Shanghai China October 2011
[27] Y Wu S S Agaian and J P Noonan ldquoSudoku associatedtwo dimensional bijections for image scramblingrdquo httparxivorgabs12075856
[28] Y YWang DWan and H Y Sheng ldquoAn encryption algorithmby scrambling image with sudoku grids matrixrdquo AdvancedMaterials Research vol 433 pp 4645ndash4650 2012
[29] J Delei B Sen and D Wenming ldquoAn image encryptionalgorithm based on Knightrsquos tour and slip encryption filterrdquo inProceedings of the International Conference on Science and Soft-ware Engineering vol 1 pp 251ndash255 Wuhan China December2008
[30] Z K Lei Q Y Sun and X X Ning ldquoImage scrambling algo-rithms based on knight-tour transform and its applicationsrdquoJournal of Chinese Computer Systems vol 5 article 044 2010
[31] K Loukhaoukha J-Y Chouinard and A Berdai ldquoA secureimage encryption algorithm based on Rubikrsquos cube principlerdquoJournal of Electrical amp Computer Engineering vol 2012 ArticleID 173931 13 pages 2012
[32] A-V Diaconu and K Loukhaoukha ldquoAn improved secureimage encryption algorithm based on Rubikrsquos cube principleand digital chaotic cipherrdquoMathematical Problems in Engineer-ing vol 2013 Article ID 848392 10 pages 2013
[33] X Wang and J Zhang ldquoAn image scrambling encryption usingchaos-controlled Poker shuffle operationrdquo in Proceedings ofthe IEEE International Symposium on Biometrics and SecurityTechnologies (ISBAST rsquo08) pp 1ndash6 Islamabad Pakistan April2008
[34] A-C Dascalescu and R Boriga ldquoA new method to improvecryptographic properties of chaotic discrete dynamical sys-temsrdquo in Proceedings of International Workshop on Informa-tion Security Theory and Practice-in Conjunction with the 7thInternational Conference for Internet Technology and SecuredTransactions pp 60ndash65 London UK December 2012
[35] A Luca A Ilyas and A Vlad ldquoGenerating random binarysequences using tent maprdquo in Proceedings of the 10th Interna-tional Symposium on Signals Circuits and Systems (ISSCS rsquo11)pp 1ndash4 Iai Romania June 2011
[36] N K Pareek V Patidar and K K Sud ldquoA random bit generatorusing chaotic mapsrdquo International Journal of Network Securityvol 10 no 1 pp 32ndash38 2010
[37] Q Zhou X Liao K-W Wong Y Hu and D Xiao ldquoTruerandom number generator based on mouse movement andchaotic hash functionrdquo Information Sciences vol 179 no 19 pp3442ndash3450 2009
[38] A-C Dascalescu R E Boriga and C Racuciu ldquoA newpseudorandom bit generator using compounded chaotic tentmapsrdquo in Proceedings of the 9th IEEE International Conferenceon Communications pp 339ndash342 Bucharest Romania June2012
[39] X-Y Wang and Y-X Xie ldquoA design of pseudo-random bitgenerator based on single chaotic systemrdquo International Journalof Modern Physics C vol 23 no 3 Article ID 1250024 2012
[40] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons and Fractals vol 40 no5 pp 2557ndash2568 2009
[41] A-C Dascalescu R E Boriga and A-V Diaconu ldquoStudy of anew chaotic dynamical system and its usage in a novel pseudo-random bit generatorrdquo Mathematical Problems in Engineeringvol 2013 Article ID 769108 10 pages 2013
[42] A-V Diaconu ldquoMultiple bitstreams generation using chaoticsequencesrdquoTheAnnals of theUniversityDunareaDe Jos of GalatiFascicle III vol 35 no 1 pp 37ndash42 2012
[43] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2004
[44] A J Menezes P C Oorschot and S A Vanstone Handbook ofApplied Cryptography CRC Press 1997
[45] httpsipiuscedudatabase
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
[46] A N Pisarchik andM Zanin ldquoImage encryption using chaoticlogistic maprdquo Physica D vol 237 no 20 pp 2638ndash2648 2008
[47] S E Borujeni andMEshghi ldquoDesign and simulation of encryp-tion system based on PRNG and Tompkins-Paige permutationalgorithm using VHDLrdquo in Proceedings of International Confer-ence on Robotics Vision Information and Signal Processing pp63ndash67 Penang Malaysia 2007
[48] D Arroyo R Rhouma G Alvarez S Li and V Fernandez ldquoOnthe security of a new image encryption scheme based on chaoticmap latticesrdquo Chaos vol 18 no 3 Article ID 033112 2008
[49] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004
[50] Y Mao G Chen and S Lian ldquoA novel fast image encryptionscheme based on 3D chaotic baker mapsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 14 no 10 pp 3613ndash3624 2004
[51] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI ran-domness tests for image encryptionrdquo Cyber Journals Multidis-ciplinary Journals in Science and Technology Journal of SelectedAreas in Telecommunications pp 31ndash38 2011
[52] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
