Research ArticleAn Experimental Realization of a Chaos-Based SecureCommunication Using Arduino Microcontrollers

Mauricio Zapateiro De la Hoz1 Leonardo Acho2 and Yolanda Vidal2

1Universidade Tecnologica Federal do Parana Avenida Alberto Carazzai 1640 86300-000 Cornelio Procopio PR Brazil2Control Dynamics and Applications Group (CoDAlab) Departament de Matematica Aplicada IIIUniversitat Politecnica de Catalunya drsquoUrgell 187 E08036 Barcelona Spain

Correspondence should be addressed to Mauricio Zapateiro De la Hoz hozutfpredubr

Received 7 May 2015 Revised 27 July 2015 Accepted 9 August 2015

Academic Editor Chengqing Li

Copyright copy 2015 Mauricio Zapateiro De la Hoz et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Security and secrecy are some of the important concerns in the communications world In the last years several encryptiontechniques have been proposed in order to improve the secrecy of the information transmitted Chaos-based encryption techniquesare being widely studied as part of the problem because of the highly unpredictable and random-look nature of the chaotic signalsIn this paper we propose a digital-based communication system that uses the logistic map which is a mathematically simple modelthat is chaotic under certain conditionsThe inputmessage signal ismodulated using a simple Deltamodulator and encrypted usinga logistic mapThe key signal is also encrypted using the same logistic map with different initial conditions In the receiver side thebinary-coded message is decrypted using the encrypted key signal that is sent through one of the communication channels Theproposed scheme is experimentally tested using Arduino shields which are simple yet powerful development kits that allows forthe implementation of the communication system for testing purposes

1 Introduction

Security and secrecy in communications are some of themost important concerns in societies nowadays With theadvent of worldwide networks and digital communicationtechniques the cryptographic techniques that once wererestricted tomilitary and state affairs are now covering severaldomains such as banks private companies medical organi-zations and so forth This has led to a very active researchfield oriented to finding optimal solutions to the problemof communications security [1ndash3] As a result numerouscryptographic techniques that seek to preserve the privacy ofthe information transmitted have been designed Chaos is thebase of many encryption and decryption techniques becausechaotic signals have a highly unpredictable and random-looknature [4]

There are basically two main approaches to designingsecure communication systems based on chaotic dynamicsanalog and digital Analog communication systems based

on chaos are possible because of the possibility of synchro-nization [5] Synchronization occurs when the output of thedriving system (master) controls the response system (slave)in such a way that they both oscillate in a synchronizedmanner On the other hand digital chaos communicationsystems do not depend on chaos synchronization at allInstead they usually use one or more chaotic maps in whichthe initial conditions and the control parameters play the roleof the secret key [6]

Several examples of chaos-based communication systemscan be found in the literature For instance Zapateiro et al[7] designed a chaotic communication system in which abinary signal is encrypted in the frequency of the sinusoidalterm of a chaotic Duffing oscillator Two chaotic signals ofthe oscillator are further encrypted with a Delta modulatorbefore they are sent through the channel In the receiver aLyapunov-based observer uses the chaotic signals for retriev-ing the sinusoidal term that contains the message A novelfrequency estimator is then used to obtain the binary signal

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 123080 10 pageshttpdxdoiorg1011552015123080

2 The Scientific World Journal

Furthermore in a new proposal Zapateiro De la Hoz et al[8] investigated a modified Chua chaotic oscillator in whichthe nonlinear term of the original oscillator was changed for asmooth and bounded function that allows for easier analysisand synchronization with other oscillators An applicationto secure communications using the modified oscillatorwas developed and its performance evaluated by numericalsimulations Hammami [9] proposed an image cryptosystemthat makes use of hyperchaotic systems Synchronizationwas achieved by assuming some structural assumptions ofthe master system and using some aggregation techniquesassociated with the arrow form matrix Fallahi and Leung[10] developed a chaotic communication system based ona chaos multiplication modulator that encrypts the signalThe chaotic signal is generated by using the Genesio-Tesichaotic system The authors also prove that the systemsecurity could not be broken with the existing methods atthat time Liu and Sun [11] propose a new design of chaoticcryptosystems in which they use high dimensional chaoticmaps along with some cryptography techniques to achievea high security level The high dimensionality of the mapleads to a high complexity and effective byte confusion anddiffusion of the output ciphertext at the time that the smallkey space problem is overcome Pareek et al [12] designedan image encryption scheme in which two logistic mapsare used along with an 80-bit key to encryptdecrypt theimages Eight different types of operation are used to encryptthe pixels of an image the type of operation is chosenaccording to the outcome of the logistic maps This securecommunication scheme was criptanalyzed in detail in Li etal [13] Lee et al [14] proposed a chaotic cipher stream a newscheme for generating pseudorandom numbers based on thecomposition of chaotic maps The method consists of usingone chaotic map to generate a sequence of pseudorandombytes and then apply some permutation on them usinganother chaotic map Shyamsunder and Kaliyaperumal [15]incorporate the concept of modular arithmetic and chaoticmaps for image encryption and decryption Zhang et al [16]propose a simple but secure chaotic cipher by improving thefamiliar permutation-diffusion structure

Numerous works can be found in the literature that usethe logistic map for improving security in communicationsThe logistic map is a nonlinear discrete map originally usedfor modeling population growth of different species as wellas economic and political phenomena [17ndash19] Howeverunder certain conditions it exhibits a chaotic behavior [20]This characteristic has been exploited in cryptography eversince For example Murillo-Escobar et al [21] presented asymmetric text cipher in which they used a 128-bit secret keytwo logistic maps with optimized pseudorandom sequencesplain text characteristics and one permutation diffusionsround Volos et al [22] presented a chaotic random bitgenerator and implemented it in an Arduino board Themicrocontroller runs side by side two logistic maps workingin different chaotic regimes due to the different initial con-ditions and system parameters Statistical tests were carriedout to prove security against intruders Pande and Zambreno[23] presented another experimental realization of a chaoticencryption scheme this time using a Xilin Virtex 6 FPGA

They implemented a modified logistic map that improvesthe performance of the logistic map in terms of Lyapunovexponent and uniformity of the bifurcation diagram Otherproposals can be found in Lawrance and Wolff [24] Chang[25] and Singh and Sinha [26]

In this paper we present a digital chaos communica-tion system in which the logistic map is used to encryptthe message and key of the transmission A simple Deltamodulator is used along with one of the chaotic maps toencrypt the message The Delta modulation technique is oneof the most simple and robust methods of analog-to-digital(ADC) schemes requiring serial digital communicationsof analog signals [27] In this work the transmitter andreceiver are implemented in low cost small but powerfulmicrocontroller boards Arduino Uno R3 [28] The Arduinotransmitter receives a message which is analog in nature andencrypts it using a logisticmap and theDeltamodulatorThenthe Arduino receiver decrypts the message and converts itto digital form which corresponds to the Delta-modulatedsignal In order to obtain the analog version of the messagesignal an analog circuitry performs the demodulation andretrieves the message

This paper is organized as follows Section 2 describes theproblem to be treated and a scheme of the proposed solutionSection 3 is a brief introduction to the logistic map and itsapplications to secure communications Section 4 presentsthe details of the implementation of the proposed techniqueFinally the conclusions are presented in Section 5

2 Problem Statement

The objective of this paper is to design and implement acommunication system to transmit a message 119898(119905) betweentwo points The goal is to use the logistic map to encrypt theinformation as a security means The proposed communica-tion system scheme is shown in Figure 1 and it consists of thefollowing blocks

(i) Arduino TransmitterThis is the core of the transmit-ter The Arduino board will take the message 119898(119905)through one of its analog input ports The Arduinowill sample the analog input message 119898(119905) andconvert it to the sampled signal 119898(119896) 119896 = 119899119879 119879 isthe sampling time 119899 = 0 1 2 This signal is thenencrypted by using a logistic map and a simple Deltamodulator Afterwards a key signal 119904(119896) is generatedin order to decrypt the message in the receiver Thiskey signal is further encrypted using a second logisticmap As a result the Arduino transmitter generatesthree outputs the first one is the encrypted message119898119890(119896) the second one is the encrypted key signal119904119890(119896) and the third one is an auxiliary key signal1199041(119896) that is used for decryption purposes

(ii) Channels Three wired channels are used to send theencrypted message and key signals

(iii) Arduino ReceiverThis is one of the twomain blocks inthe receiver side It takes the signals119898

119890(119896) 119904119890(119896) and

1199041(119896) to decrypt the Delta-modulated signal before

The Scientific World Journal 3

Arduino transmitter

Logmap 1 Decryption

Analog electronics

Deltademod

Arduino receiver

10-bitADC

Deltamodulator

Logmap 2 Encr 2

Encr 1m(k) mb(k)m(t)

x1(k)

me(k)

x2(k) s1(k)

se(k)

md(k) mr(t)s(k)

Figure 1 Block diagram of the communication system

it is converted into its analog form The output is adigital signal119898

119889(119896) which corresponds to the signal

119898119887(119896)

(iv) Delta Demodulator This is the second block in thereceiver It is a Delta demodulator consisting of anintegrator a filter and some amplifiers to retrievethe original message Its output is a signal 119898

119903(119905) that

approximates the original signal119898(119905)

The details of these blocks will be outlined in the follow-ing sections of this chapter

3 The Logistic Map

The logistic map has its origins in the works by the Belgianmathematician Pierre-Francois Verhulst in the first half ofthe 18th century [29 30] Verhulst published in 1845 and1847 two articles on how the population growth could bemathematically modeled He called this model the logisticcurve [31 32] and it is the continuous time version of whatnowadays is known as the logistic map

The logistic map the discrete-time version of Verhulstrsquoslogistic model is chaotic under certain conditions Its equa-tion is

119909119894+1= 119903119909119894(1 minus 119909

119894) 0 le 119909

119894le 1 (1)

where 119903 is a constant parameter Figure 2 is the bifurcationdiagram of the logistic map created by varying the parameter119903 from 25 to 40

As can be seen in the bifurcation diagram there aredifferent regions that depend on the value of 119903 It is ofparticular interest when 119903 = 3 because there it begins theperiod doubling that leads to the chaotic dynamics when119903 asymp 35699 until 119903 = 40 Figure 3 shows the Lyapunovexponent of the logistic map as 119903 is varied from 25 to 40 Itcan be seen that the Lyapunov exponent 120582 becomes positivefor values of approximately greater than 356which is a strongindicator of chaos [33]

In the next sections we will use a logistic map aspart of an encryptiondecryption scheme for transmittinginformation We will explain the details of the prototype ofthis communication system which is implemented on twoArduino Uno boards

4 Experimental Implementation

41 Description of the Communication System The com-munication system implemented in this work consists of

25 3 35 40

02

04

06

08

1

rxn

Figure 2 Logistic map bifurcation diagram

25 3 35 4

0

1

r

120582minus1

minus2

minus3

minus4

Figure 3 Logistic map Lyapunov exponent

a transmitter and a receiver whose cores are the ArduinoUnoR3 microcontroller boards [28] These are low cost simplebut powerful microcontrollers based on the ATmega328 chipThey have 14 digital inputoutput pins (6 of them can beused as PWM outputs) 6 analog inputs a 16MHz crystaloscillator a USB connection and a reset button They can beprogrammed using a language similar to C++ called Wiring

The flow diagram of the programs executed by eachArduino is shown in Figures 4 and 5 in order to facilitate thedescription of the communication system algorithms

The communication begins when a message119898(119905) gener-ated by a function generator and is sent to the analog inputA0 of the Arduino transmitter Arduino analog inputs onlyaccept unipolar signals in the range from 0V to 5V Anembedded 10-bit ADC converts the input signal from analogto digital at a maximum rate of 10000 samples per secondSince the output of theADC is a value between 0 and 1023 (theADC resolution) an internal operation to bring it back to therange from 0V to 5V is executed The result is the sampledmessage signal119898(119896)

4 The Scientific World Journal

mb(k) = true

aux = 5

Start TX

Initialize parametersh = 01 aux = 5 r = 39018

Initialize variables

x1 = 01 x2 = 05 xn = 0

Configure Arduino ports

D2 D4 D7 out

Read port A0

m(k) minus xn gt 0

xn(k + 1) = xn(k) + h lowast aux

x2(k) = r lowast x2(k) lowast (1 minus x2(k))

x1(k) = r lowast x1(k) lowast (1 minus x1(k))

x1(k) gt 05

me(k) = mb(k)

x2(k) lt 01

s1(k) = s(k)

s1(k) =s(k)

s(k) = true

se(k) = s1(k) and s2(k)

or s1(k) and s2(k)

s(k) = false

s2(k) = true

se(k) = false

mb(k) = falseD2 se(k)D4 me(k)

D7 s1(k)

Write to ports

1

1

2

2

No

No

No

Yes

Yes

Yes

aux = minus5

me(k) = mb(k)

m(k) m(t)

Figure 4 Flow diagram of the transmitter Arduino codes

The Scientific World Journal 5

Start RX

Configure Arduino portsD2 D4 D7 in

D3 out

Read portsD2 se(k)

D4 me(k)

D7 s1(k)

s2(k) = s1(k) and se(k)

or s1(k) and se(k)

s2(k) = trueNo

No

Yes

Yes

s(k) = s1(k)

s(k) = s1(k)

s(k) = trueWrite to ports

Write to portsD3 me(k)

1

1

D3 me(k)

Figure 5 Flow diagram of the receiver Arduino codes

The next step is the Delta modulation This kind ofmodulation can be viewed as an 1-bit ADC conversionscheme since it generates one output bit per input sampleThe scheme of the Delta modulation is shown in Figure 6 Itconsists of a comparator in the forward path and an integratorin the feedback path of a simple control loop The inputs ofthe comparator are the signal to be modulated119898(119896) and theoutput of the integrator 119909

119899(119896) As a result the modulated

output 119898119887(119896) is either true (high) or false (low) at any

given time as shown in Figure 7 In this figure we see aninput signal and the integral of the expression 119898(119896) minus 119909

119887(119896)

Clock

Delta modulatorTransmitter (Arduino)

Delta demodulatorReceiver (analog elect)

+

minus

int

intmb(k) mb(k) m(k)m(k)

xn(k)

Figure 6 Diagram of the simple Delta modulator

0 002 004 006 008 010246

Inpu

tIn signalRec signal

Time

(a)

0 002 004 006 008 01

012

Out

put

Time

minus1

(b)

Figure 7 Delta modulation example of a sine input signal (a)Input signal and reconstructed signal comparison (b) Modulatedoutput

For instance if 119898119887(119896) is ramping up and its output is less

than the input the integrator output will continue rampingup otherwise it will ramp down The signal 119898

119887(119896) is the

differential of the input and thus it can be reconstructed inthe receiver by integrating it In this work the integral signal119909119899(119896) is digitally generated by the Arduino program On

the other hand the reconstructing integrator of the receiveris implemented with analog electronics as will be explainedlater A full description of the Delta modulation techniquecan be found in Taylor [27]

After one bit from the Delta modulator is obtained thenext step is the message encryption In order to do so twologistic maps are called to generate two values 119909

1(119896) and

1199092(119896) The logistic maps have different initial conditions

that is 1199091(0) = 119909

2(0) Firstly the message is coded with

a value true or false that is assigned depending on thevalue 119909

1(119896) of the first chaotic map as can be seen in

Algorithm 1 (Part 1) where 119898119890(119896) is the encrypted message

and 119904(119896) is the key necessary to retrieve119898119890(119896)

In order to increase the security of the system the key119904(119896) is further encrypted following the same scheme It is

6 The Scientific World Journal

Part 1(1) if 119909

1(119896) gt 05 then

(2) 119898119890(119896) =119898

119887(119896)

(3) 119904(119896) = true(4) else(5) 119898

119890(119896) = 119898

119887(119896) ⊳Symbol means boolean negation

(6) 119904(119896) = false(7) end ifPart 2(8) if 119909

2(119896) lt 01 then

(9) 1199041(119896) = 119904(119896) ⊳Symbol means boolean negation

(10) 1199042(119896) = true

(11) else(12) 119904

1(119896) = 119904(119896)

(13) 1199042(119896) = false

(14) end ifPart 3(15) 119904119890(119896) = (119904

1(119896) AND 119904

2(119896)) OR (119904

1(119896) AND 119904

2(119896))

Algorithm 1

done by assigning it a value true or false that depends onthe second chaotic map value 119909

2(119896) as shown in Algorithm 1

(Part 2) where 1199041(119896) and 119904

2(119896) are auxiliary signals that are

used for encrypting and decrypting the key signal 119904(119896)The key is then finally encrypted by applying the

XOR function to the variables 1199041(119896) and 119904

2(119896) to yield

Algorithm 1 (part 3)The signals 119904

119890(119896)119898

119890(119896) and 119904

1(119896) are sent to the receiver

through digital outputs D2 D4 and D7 respectivelyIn the receiver the signals 119904

119890(119896) 119898

119890(119896) and 119904

1(119896) go

directly to the Arduino inputs D2 D4 and D7 respectivelyTheflowdiagramof the receiver program is shown in Figure 5as well The first step in decrypting the message is thedecryption of the key signal 119904

119890(119896) This is done by applying

the boolean formula that reverts the encryptionThe formulato calculate 119904

2(119896) given 119904

119890(119896) and 119904

1(119896) is obtained as follows

Recall that in the transmitter 119904119890(119896) is obtained by using the

XOR function

119904119890(119896) = 119904

1015840

1(119896) sdot 1199042(119896) + 119904

1(119896) sdot 119904

1015840

2(119896) (2)

where (sdot)1015840 is the complement operation of the correspondinglogic variable The truth table of the function in (2) is shownin Table 1 Thus given 119904

1(119896) and 119904

119890(119896) for obtaining the signal

1199042(119896) would result in the truth Table 2The Karnaugh maps technique [34] was used to find

the desired simplified expression for 1199042(119896) It is a pictorial

method in which the truth table of the boolean functionto be simplified is represented in a bidimensional formThe boolean variables are arranged according to the Graycode The terms of the simplified expression are found bygrouping 1s or 0s in an optimal way and therefore eliminatingunnecessary variables As a result the following booleanexpression for 119904

2(119896) is obtained

1199042(119896) = 119904

1015840

1(119896) sdot 119904119890(119896) + 119904

1(119896) sdot 119904

1015840

119890(119896) (3)

Table 1 Truth table for 119904119890(119896)

1199041(119896) 119904

2(119896) 119904

119890(119896)

0 0 00 1 11 0 11 1 0

Table 2 Truth table for 1199042(119896)

1199041(119896) 119904

119890(119896) 119904

2(119896)

0 0 00 1 11 0 01 1 1

Once 1199042(119896) is retrieved the signal 119904(119896) is obtained with

Algorithm 2 (part 1)The signal 119898

119890(119896) is finally decrypted by analyzing the

value of 119904(119896) (see Algorithm 2 (part 2)) where 119898119889(119896) is the

decrypted signal The output 119898119889(119896) is sent to the output pin

D3 and it goes directly to theDelta demodulator realizedwithanalog electronics using operational amplifiers

As shown in Figure 6 the Delta demodulation consists ofan integrator The signal is passed through different stages asshown in the circuit diagram of Figure 8The circuit has threemain blocksThe first one composed of the amplifiers U1 andU2 is a unipolar to bipolar converter Recall that the Arduinoinputsmust be unipolar so in the case that the original signalsare bipolar they must be recovered to its original form at theoutput of the Arduino Thus the signal 119898(119896) isin [0 5]V isconverted to a signal 119898(119905) isin [minus25 25]V The second blockis composed of amplifiers U3 and U4 They are designed tocompute the integral of the input signal It consists of anintegrator that performs the Delta demodulation (U4) and

The Scientific World Journal 7

1k

1k1k

m(t) me(k)

me(k)

s(k)

s(k)

mb(k)

mr(t)

+

+

+

+ +

+ +

+

+

+

+

minus

minus

minus

minus

minus minus

minus

TL082TL082

TL082

TL082 TL082

TL082 TL082

minus5V

minus5V minus5V

minus5V

minus5V minus5V

minus5V

minus5V

47120583F

47120583F

1k

1k

Vy

Vy

Vx

Vx

U6 U7

5V5V

5V 5V 5V

5V5V

U3

U2U1

U4 U5

01 120583F

10k 10k

10k

10k 10k

10k

10k

Arduino RX

Arduino TX

and+

and+

andminus

and+

andminus

andminus

and+

andminus

and+

andminus

and+

andminus

and+

andminus

22 k

33 120583F

Figure 8 Circuit diagram of the analog electronics in the receiver

an inverter amplifier (U3) to adjust the quality of its outputThese signals are finally sent through a low-pass filter anamplifier and an inverter (amplifiers U5ndashU7) to get the final119898119903(119905) which should be approximately equal to119898(119905)

42 Experimental Results For the experiments the logisticmaps were implemented with 119903 = 39018 and initial con-ditions 119909

1(0) = 01 and 119909

2(0) = 05 As an example the

sequence of numbers generated by the logistic map when119909(0) = 05 is shown in Figure 9 Each loop of the transmitteralgorithm is executed by the Arduino microcontroller in210 120583s approximately while each receiver loop is executedin 25 120583s approximately This means that the message signalbandwidth should be at most 500Hz approximately in orderto be well retrieved in the receiver

Figures 10 to 13 are screenshots of the oscilloscopecorresponding to the first experiment In this case a 125Hzsinewave 5 V peak-to-peak amplitude was used as amessagesignal In Figure 10 we see a comparison of the sent message119898(119905) (in blue) and the retrieved message 119898

119903(119905) (in yellow)

Figure 11 compares the sent message 119898(119905) (in blue) andthe encrypted message signal 119898

119890(119896) (in yellow) Figure 12

is a comparison on the sent message 119898(119905) (in blue) andthe encrypted key signal 119904

119890(119896) (in yellow) Finally Figure 13

compares the sent message (in blue) and the auxiliary signal1199041(119896) (in yellow)In subsequent experiments different frequencies and

waveforms were tested Figure 14 shows a 125Hz triangularwave message (in blue) and its retrieved version (in yellow)Figure 15 compares a 70Hz sine wave message (in blue) and

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

The Scientific World Journal 3

Arduino transmitter

Logmap 1 Decryption

Analog electronics

Deltademod

Arduino receiver

10-bitADC

Deltamodulator

Logmap 2 Encr 2

Encr 1m(k) mb(k)m(t)

x1(k)

me(k)

x2(k) s1(k)

se(k)

md(k) mr(t)s(k)

Figure 1 Block diagram of the communication system

it is converted into its analog form The output is adigital signal119898

119889(119896) which corresponds to the signal

119898119887(119896)

(iv) Delta Demodulator This is the second block in thereceiver It is a Delta demodulator consisting of anintegrator a filter and some amplifiers to retrievethe original message Its output is a signal 119898

119903(119905) that

approximates the original signal119898(119905)

The details of these blocks will be outlined in the follow-ing sections of this chapter

3 The Logistic Map

The logistic map has its origins in the works by the Belgianmathematician Pierre-Francois Verhulst in the first half ofthe 18th century [29 30] Verhulst published in 1845 and1847 two articles on how the population growth could bemathematically modeled He called this model the logisticcurve [31 32] and it is the continuous time version of whatnowadays is known as the logistic map

The logistic map the discrete-time version of Verhulstrsquoslogistic model is chaotic under certain conditions Its equa-tion is

119909119894+1= 119903119909119894(1 minus 119909

119894) 0 le 119909

119894le 1 (1)

where 119903 is a constant parameter Figure 2 is the bifurcationdiagram of the logistic map created by varying the parameter119903 from 25 to 40

As can be seen in the bifurcation diagram there aredifferent regions that depend on the value of 119903 It is ofparticular interest when 119903 = 3 because there it begins theperiod doubling that leads to the chaotic dynamics when119903 asymp 35699 until 119903 = 40 Figure 3 shows the Lyapunovexponent of the logistic map as 119903 is varied from 25 to 40 Itcan be seen that the Lyapunov exponent 120582 becomes positivefor values of approximately greater than 356which is a strongindicator of chaos [33]

In the next sections we will use a logistic map aspart of an encryptiondecryption scheme for transmittinginformation We will explain the details of the prototype ofthis communication system which is implemented on twoArduino Uno boards

4 Experimental Implementation

41 Description of the Communication System The com-munication system implemented in this work consists of

25 3 35 40

02

04

06

08

1

rxn

Figure 2 Logistic map bifurcation diagram

25 3 35 4

0

1

r

120582minus1

minus2

minus3

minus4

Figure 3 Logistic map Lyapunov exponent

a transmitter and a receiver whose cores are the ArduinoUnoR3 microcontroller boards [28] These are low cost simplebut powerful microcontrollers based on the ATmega328 chipThey have 14 digital inputoutput pins (6 of them can beused as PWM outputs) 6 analog inputs a 16MHz crystaloscillator a USB connection and a reset button They can beprogrammed using a language similar to C++ called Wiring

The flow diagram of the programs executed by eachArduino is shown in Figures 4 and 5 in order to facilitate thedescription of the communication system algorithms

The communication begins when a message119898(119905) gener-ated by a function generator and is sent to the analog inputA0 of the Arduino transmitter Arduino analog inputs onlyaccept unipolar signals in the range from 0V to 5V Anembedded 10-bit ADC converts the input signal from analogto digital at a maximum rate of 10000 samples per secondSince the output of theADC is a value between 0 and 1023 (theADC resolution) an internal operation to bring it back to therange from 0V to 5V is executed The result is the sampledmessage signal119898(119896)

4 The Scientific World Journal

mb(k) = true

aux = 5

Start TX

Initialize parametersh = 01 aux = 5 r = 39018

Initialize variables

x1 = 01 x2 = 05 xn = 0

Configure Arduino ports

D2 D4 D7 out

Read port A0

m(k) minus xn gt 0

xn(k + 1) = xn(k) + h lowast aux

x2(k) = r lowast x2(k) lowast (1 minus x2(k))

x1(k) = r lowast x1(k) lowast (1 minus x1(k))

x1(k) gt 05

me(k) = mb(k)

x2(k) lt 01

s1(k) = s(k)

s1(k) =s(k)

s(k) = true

se(k) = s1(k) and s2(k)

or s1(k) and s2(k)

s(k) = false

s2(k) = true

se(k) = false

mb(k) = falseD2 se(k)D4 me(k)

D7 s1(k)

Write to ports

1

1

2

2

No

No

No

Yes

Yes

Yes

aux = minus5

me(k) = mb(k)

m(k) m(t)

Figure 4 Flow diagram of the transmitter Arduino codes

The Scientific World Journal 5

Start RX

Configure Arduino portsD2 D4 D7 in

D3 out

Read portsD2 se(k)

D4 me(k)

D7 s1(k)

s2(k) = s1(k) and se(k)

or s1(k) and se(k)

s2(k) = trueNo

No

Yes

Yes

s(k) = s1(k)

s(k) = s1(k)

s(k) = trueWrite to ports

Write to portsD3 me(k)

1

1

D3 me(k)

Figure 5 Flow diagram of the receiver Arduino codes

The next step is the Delta modulation This kind ofmodulation can be viewed as an 1-bit ADC conversionscheme since it generates one output bit per input sampleThe scheme of the Delta modulation is shown in Figure 6 Itconsists of a comparator in the forward path and an integratorin the feedback path of a simple control loop The inputs ofthe comparator are the signal to be modulated119898(119896) and theoutput of the integrator 119909

119899(119896) As a result the modulated

output 119898119887(119896) is either true (high) or false (low) at any

given time as shown in Figure 7 In this figure we see aninput signal and the integral of the expression 119898(119896) minus 119909

119887(119896)

Clock

Delta modulatorTransmitter (Arduino)

Delta demodulatorReceiver (analog elect)

+

minus

int

intmb(k) mb(k) m(k)m(k)

xn(k)

Figure 6 Diagram of the simple Delta modulator

0 002 004 006 008 010246

Inpu

tIn signalRec signal

Time

(a)

0 002 004 006 008 01

012

Out

put

Time

minus1

(b)

Figure 7 Delta modulation example of a sine input signal (a)Input signal and reconstructed signal comparison (b) Modulatedoutput

For instance if 119898119887(119896) is ramping up and its output is less

than the input the integrator output will continue rampingup otherwise it will ramp down The signal 119898

119887(119896) is the

differential of the input and thus it can be reconstructed inthe receiver by integrating it In this work the integral signal119909119899(119896) is digitally generated by the Arduino program On

the other hand the reconstructing integrator of the receiveris implemented with analog electronics as will be explainedlater A full description of the Delta modulation techniquecan be found in Taylor [27]

After one bit from the Delta modulator is obtained thenext step is the message encryption In order to do so twologistic maps are called to generate two values 119909

1(119896) and

1199092(119896) The logistic maps have different initial conditions

that is 1199091(0) = 119909

2(0) Firstly the message is coded with

a value true or false that is assigned depending on thevalue 119909

1(119896) of the first chaotic map as can be seen in

Algorithm 1 (Part 1) where 119898119890(119896) is the encrypted message

and 119904(119896) is the key necessary to retrieve119898119890(119896)

In order to increase the security of the system the key119904(119896) is further encrypted following the same scheme It is

6 The Scientific World Journal

Part 1(1) if 119909

1(119896) gt 05 then

(2) 119898119890(119896) =119898

119887(119896)

(3) 119904(119896) = true(4) else(5) 119898

119890(119896) = 119898

119887(119896) ⊳Symbol means boolean negation

(6) 119904(119896) = false(7) end ifPart 2(8) if 119909

2(119896) lt 01 then

(9) 1199041(119896) = 119904(119896) ⊳Symbol means boolean negation

(10) 1199042(119896) = true

(11) else(12) 119904

1(119896) = 119904(119896)

(13) 1199042(119896) = false

(14) end ifPart 3(15) 119904119890(119896) = (119904

1(119896) AND 119904

2(119896)) OR (119904

1(119896) AND 119904

2(119896))

Algorithm 1

done by assigning it a value true or false that depends onthe second chaotic map value 119909

2(119896) as shown in Algorithm 1

(Part 2) where 1199041(119896) and 119904

2(119896) are auxiliary signals that are

used for encrypting and decrypting the key signal 119904(119896)The key is then finally encrypted by applying the

XOR function to the variables 1199041(119896) and 119904

2(119896) to yield

Algorithm 1 (part 3)The signals 119904

119890(119896)119898

119890(119896) and 119904

1(119896) are sent to the receiver

through digital outputs D2 D4 and D7 respectivelyIn the receiver the signals 119904

119890(119896) 119898

119890(119896) and 119904

1(119896) go

directly to the Arduino inputs D2 D4 and D7 respectivelyTheflowdiagramof the receiver program is shown in Figure 5as well The first step in decrypting the message is thedecryption of the key signal 119904

119890(119896) This is done by applying

the boolean formula that reverts the encryptionThe formulato calculate 119904

2(119896) given 119904

119890(119896) and 119904

1(119896) is obtained as follows

Recall that in the transmitter 119904119890(119896) is obtained by using the

XOR function

119904119890(119896) = 119904

1015840

1(119896) sdot 1199042(119896) + 119904

1(119896) sdot 119904

1015840

2(119896) (2)

where (sdot)1015840 is the complement operation of the correspondinglogic variable The truth table of the function in (2) is shownin Table 1 Thus given 119904

1(119896) and 119904

119890(119896) for obtaining the signal

1199042(119896) would result in the truth Table 2The Karnaugh maps technique [34] was used to find

the desired simplified expression for 1199042(119896) It is a pictorial

method in which the truth table of the boolean functionto be simplified is represented in a bidimensional formThe boolean variables are arranged according to the Graycode The terms of the simplified expression are found bygrouping 1s or 0s in an optimal way and therefore eliminatingunnecessary variables As a result the following booleanexpression for 119904

2(119896) is obtained

1199042(119896) = 119904

1015840

1(119896) sdot 119904119890(119896) + 119904

1(119896) sdot 119904

1015840

119890(119896) (3)

Table 1 Truth table for 119904119890(119896)

1199041(119896) 119904

2(119896) 119904

119890(119896)

0 0 00 1 11 0 11 1 0

Table 2 Truth table for 1199042(119896)

1199041(119896) 119904

119890(119896) 119904

2(119896)

0 0 00 1 11 0 01 1 1

Once 1199042(119896) is retrieved the signal 119904(119896) is obtained with

Algorithm 2 (part 1)The signal 119898

119890(119896) is finally decrypted by analyzing the

value of 119904(119896) (see Algorithm 2 (part 2)) where 119898119889(119896) is the

decrypted signal The output 119898119889(119896) is sent to the output pin

D3 and it goes directly to theDelta demodulator realizedwithanalog electronics using operational amplifiers

As shown in Figure 6 the Delta demodulation consists ofan integrator The signal is passed through different stages asshown in the circuit diagram of Figure 8The circuit has threemain blocksThe first one composed of the amplifiers U1 andU2 is a unipolar to bipolar converter Recall that the Arduinoinputsmust be unipolar so in the case that the original signalsare bipolar they must be recovered to its original form at theoutput of the Arduino Thus the signal 119898(119896) isin [0 5]V isconverted to a signal 119898(119905) isin [minus25 25]V The second blockis composed of amplifiers U3 and U4 They are designed tocompute the integral of the input signal It consists of anintegrator that performs the Delta demodulation (U4) and

The Scientific World Journal 7

1k

1k1k

m(t) me(k)

me(k)

s(k)

s(k)

mb(k)

mr(t)

+

+

+

+ +

+ +

+

+

+

+

minus

minus

minus

minus

minus minus

minus

TL082TL082

TL082

TL082 TL082

TL082 TL082

minus5V

minus5V minus5V

minus5V

minus5V minus5V

minus5V

minus5V

47120583F

47120583F

1k

1k

Vy

Vy

Vx

Vx

U6 U7

5V5V

5V 5V 5V

5V5V

U3

U2U1

U4 U5

01 120583F

10k 10k

10k

10k 10k

10k

10k

Arduino RX

Arduino TX

and+

and+

andminus

and+

andminus

andminus

and+

andminus

and+

andminus

and+

andminus

and+

andminus

22 k

33 120583F

Figure 8 Circuit diagram of the analog electronics in the receiver

an inverter amplifier (U3) to adjust the quality of its outputThese signals are finally sent through a low-pass filter anamplifier and an inverter (amplifiers U5ndashU7) to get the final119898119903(119905) which should be approximately equal to119898(119905)

42 Experimental Results For the experiments the logisticmaps were implemented with 119903 = 39018 and initial con-ditions 119909

1(0) = 01 and 119909

2(0) = 05 As an example the

sequence of numbers generated by the logistic map when119909(0) = 05 is shown in Figure 9 Each loop of the transmitteralgorithm is executed by the Arduino microcontroller in210 120583s approximately while each receiver loop is executedin 25 120583s approximately This means that the message signalbandwidth should be at most 500Hz approximately in orderto be well retrieved in the receiver

Figures 10 to 13 are screenshots of the oscilloscopecorresponding to the first experiment In this case a 125Hzsinewave 5 V peak-to-peak amplitude was used as amessagesignal In Figure 10 we see a comparison of the sent message119898(119905) (in blue) and the retrieved message 119898

119903(119905) (in yellow)

Figure 11 compares the sent message 119898(119905) (in blue) andthe encrypted message signal 119898

119890(119896) (in yellow) Figure 12

is a comparison on the sent message 119898(119905) (in blue) andthe encrypted key signal 119904

119890(119896) (in yellow) Finally Figure 13

compares the sent message (in blue) and the auxiliary signal1199041(119896) (in yellow)In subsequent experiments different frequencies and

waveforms were tested Figure 14 shows a 125Hz triangularwave message (in blue) and its retrieved version (in yellow)Figure 15 compares a 70Hz sine wave message (in blue) and

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

4 The Scientific World Journal

mb(k) = true

aux = 5

Start TX

Initialize parametersh = 01 aux = 5 r = 39018

Initialize variables

x1 = 01 x2 = 05 xn = 0

Configure Arduino ports

D2 D4 D7 out

Read port A0

m(k) minus xn gt 0

xn(k + 1) = xn(k) + h lowast aux

x2(k) = r lowast x2(k) lowast (1 minus x2(k))

x1(k) = r lowast x1(k) lowast (1 minus x1(k))

x1(k) gt 05

me(k) = mb(k)

x2(k) lt 01

s1(k) = s(k)

s1(k) =s(k)

s(k) = true

se(k) = s1(k) and s2(k)

or s1(k) and s2(k)

s(k) = false

s2(k) = true

se(k) = false

mb(k) = falseD2 se(k)D4 me(k)

D7 s1(k)

Write to ports

1

1

2

2

No

No

No

Yes

Yes

Yes

aux = minus5

me(k) = mb(k)

m(k) m(t)

Figure 4 Flow diagram of the transmitter Arduino codes

The Scientific World Journal 5

Start RX

Configure Arduino portsD2 D4 D7 in

D3 out

Read portsD2 se(k)

D4 me(k)

D7 s1(k)

s2(k) = s1(k) and se(k)

or s1(k) and se(k)

s2(k) = trueNo

No

Yes

Yes

s(k) = s1(k)

s(k) = s1(k)

s(k) = trueWrite to ports

Write to portsD3 me(k)

1

1

D3 me(k)

Figure 5 Flow diagram of the receiver Arduino codes

The next step is the Delta modulation This kind ofmodulation can be viewed as an 1-bit ADC conversionscheme since it generates one output bit per input sampleThe scheme of the Delta modulation is shown in Figure 6 Itconsists of a comparator in the forward path and an integratorin the feedback path of a simple control loop The inputs ofthe comparator are the signal to be modulated119898(119896) and theoutput of the integrator 119909

119899(119896) As a result the modulated

output 119898119887(119896) is either true (high) or false (low) at any

given time as shown in Figure 7 In this figure we see aninput signal and the integral of the expression 119898(119896) minus 119909

119887(119896)

Clock

Delta modulatorTransmitter (Arduino)

Delta demodulatorReceiver (analog elect)

+

minus

int

intmb(k) mb(k) m(k)m(k)

xn(k)

Figure 6 Diagram of the simple Delta modulator

0 002 004 006 008 010246

Inpu

tIn signalRec signal

Time

(a)

0 002 004 006 008 01

012

Out

put

Time

minus1

(b)

Figure 7 Delta modulation example of a sine input signal (a)Input signal and reconstructed signal comparison (b) Modulatedoutput

For instance if 119898119887(119896) is ramping up and its output is less

than the input the integrator output will continue rampingup otherwise it will ramp down The signal 119898

119887(119896) is the

differential of the input and thus it can be reconstructed inthe receiver by integrating it In this work the integral signal119909119899(119896) is digitally generated by the Arduino program On

the other hand the reconstructing integrator of the receiveris implemented with analog electronics as will be explainedlater A full description of the Delta modulation techniquecan be found in Taylor [27]

After one bit from the Delta modulator is obtained thenext step is the message encryption In order to do so twologistic maps are called to generate two values 119909

1(119896) and

1199092(119896) The logistic maps have different initial conditions

that is 1199091(0) = 119909

2(0) Firstly the message is coded with

a value true or false that is assigned depending on thevalue 119909

1(119896) of the first chaotic map as can be seen in

Algorithm 1 (Part 1) where 119898119890(119896) is the encrypted message

and 119904(119896) is the key necessary to retrieve119898119890(119896)

In order to increase the security of the system the key119904(119896) is further encrypted following the same scheme It is

6 The Scientific World Journal

Part 1(1) if 119909

1(119896) gt 05 then

(2) 119898119890(119896) =119898

119887(119896)

(3) 119904(119896) = true(4) else(5) 119898

119890(119896) = 119898

119887(119896) ⊳Symbol means boolean negation

(6) 119904(119896) = false(7) end ifPart 2(8) if 119909

2(119896) lt 01 then

(9) 1199041(119896) = 119904(119896) ⊳Symbol means boolean negation

(10) 1199042(119896) = true

(11) else(12) 119904

1(119896) = 119904(119896)

(13) 1199042(119896) = false

(14) end ifPart 3(15) 119904119890(119896) = (119904

1(119896) AND 119904

2(119896)) OR (119904

1(119896) AND 119904

2(119896))

Algorithm 1

done by assigning it a value true or false that depends onthe second chaotic map value 119909

2(119896) as shown in Algorithm 1

(Part 2) where 1199041(119896) and 119904

2(119896) are auxiliary signals that are

used for encrypting and decrypting the key signal 119904(119896)The key is then finally encrypted by applying the

XOR function to the variables 1199041(119896) and 119904

2(119896) to yield

Algorithm 1 (part 3)The signals 119904

119890(119896)119898

119890(119896) and 119904

1(119896) are sent to the receiver

through digital outputs D2 D4 and D7 respectivelyIn the receiver the signals 119904

119890(119896) 119898

119890(119896) and 119904

1(119896) go

directly to the Arduino inputs D2 D4 and D7 respectivelyTheflowdiagramof the receiver program is shown in Figure 5as well The first step in decrypting the message is thedecryption of the key signal 119904

119890(119896) This is done by applying

the boolean formula that reverts the encryptionThe formulato calculate 119904

2(119896) given 119904

119890(119896) and 119904

1(119896) is obtained as follows

Recall that in the transmitter 119904119890(119896) is obtained by using the

XOR function

119904119890(119896) = 119904

1015840

1(119896) sdot 1199042(119896) + 119904

1(119896) sdot 119904

1015840

2(119896) (2)

where (sdot)1015840 is the complement operation of the correspondinglogic variable The truth table of the function in (2) is shownin Table 1 Thus given 119904

1(119896) and 119904

119890(119896) for obtaining the signal

1199042(119896) would result in the truth Table 2The Karnaugh maps technique [34] was used to find

the desired simplified expression for 1199042(119896) It is a pictorial

method in which the truth table of the boolean functionto be simplified is represented in a bidimensional formThe boolean variables are arranged according to the Graycode The terms of the simplified expression are found bygrouping 1s or 0s in an optimal way and therefore eliminatingunnecessary variables As a result the following booleanexpression for 119904

2(119896) is obtained

1199042(119896) = 119904

1015840

1(119896) sdot 119904119890(119896) + 119904

1(119896) sdot 119904

1015840

119890(119896) (3)

Table 1 Truth table for 119904119890(119896)

1199041(119896) 119904

2(119896) 119904

119890(119896)

0 0 00 1 11 0 11 1 0

Table 2 Truth table for 1199042(119896)

1199041(119896) 119904

119890(119896) 119904

2(119896)

0 0 00 1 11 0 01 1 1

Once 1199042(119896) is retrieved the signal 119904(119896) is obtained with

Algorithm 2 (part 1)The signal 119898

119890(119896) is finally decrypted by analyzing the

value of 119904(119896) (see Algorithm 2 (part 2)) where 119898119889(119896) is the

decrypted signal The output 119898119889(119896) is sent to the output pin

D3 and it goes directly to theDelta demodulator realizedwithanalog electronics using operational amplifiers

As shown in Figure 6 the Delta demodulation consists ofan integrator The signal is passed through different stages asshown in the circuit diagram of Figure 8The circuit has threemain blocksThe first one composed of the amplifiers U1 andU2 is a unipolar to bipolar converter Recall that the Arduinoinputsmust be unipolar so in the case that the original signalsare bipolar they must be recovered to its original form at theoutput of the Arduino Thus the signal 119898(119896) isin [0 5]V isconverted to a signal 119898(119905) isin [minus25 25]V The second blockis composed of amplifiers U3 and U4 They are designed tocompute the integral of the input signal It consists of anintegrator that performs the Delta demodulation (U4) and

The Scientific World Journal 7

1k

1k1k

m(t) me(k)

me(k)

s(k)

s(k)

mb(k)

mr(t)

+

+

+

+ +

+ +

+

+

+

+

minus

minus

minus

minus

minus minus

minus

TL082TL082

TL082

TL082 TL082

TL082 TL082

minus5V

minus5V minus5V

minus5V

minus5V minus5V

minus5V

minus5V

47120583F

47120583F

1k

1k

Vy

Vy

Vx

Vx

U6 U7

5V5V

5V 5V 5V

5V5V

U3

U2U1

U4 U5

01 120583F

10k 10k

10k

10k 10k

10k

10k

Arduino RX

Arduino TX

and+

and+

andminus

and+

andminus

andminus

and+

andminus

and+

andminus

and+

andminus

and+

andminus

22 k

33 120583F

Figure 8 Circuit diagram of the analog electronics in the receiver

an inverter amplifier (U3) to adjust the quality of its outputThese signals are finally sent through a low-pass filter anamplifier and an inverter (amplifiers U5ndashU7) to get the final119898119903(119905) which should be approximately equal to119898(119905)

42 Experimental Results For the experiments the logisticmaps were implemented with 119903 = 39018 and initial con-ditions 119909

1(0) = 01 and 119909

2(0) = 05 As an example the

sequence of numbers generated by the logistic map when119909(0) = 05 is shown in Figure 9 Each loop of the transmitteralgorithm is executed by the Arduino microcontroller in210 120583s approximately while each receiver loop is executedin 25 120583s approximately This means that the message signalbandwidth should be at most 500Hz approximately in orderto be well retrieved in the receiver

Figures 10 to 13 are screenshots of the oscilloscopecorresponding to the first experiment In this case a 125Hzsinewave 5 V peak-to-peak amplitude was used as amessagesignal In Figure 10 we see a comparison of the sent message119898(119905) (in blue) and the retrieved message 119898

119903(119905) (in yellow)

Figure 11 compares the sent message 119898(119905) (in blue) andthe encrypted message signal 119898

119890(119896) (in yellow) Figure 12

is a comparison on the sent message 119898(119905) (in blue) andthe encrypted key signal 119904

119890(119896) (in yellow) Finally Figure 13

compares the sent message (in blue) and the auxiliary signal1199041(119896) (in yellow)In subsequent experiments different frequencies and

waveforms were tested Figure 14 shows a 125Hz triangularwave message (in blue) and its retrieved version (in yellow)Figure 15 compares a 70Hz sine wave message (in blue) and

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

The Scientific World Journal 5

Start RX

Configure Arduino portsD2 D4 D7 in

D3 out

Read portsD2 se(k)

D4 me(k)

D7 s1(k)

s2(k) = s1(k) and se(k)

or s1(k) and se(k)

s2(k) = trueNo

No

Yes

Yes

s(k) = s1(k)

s(k) = s1(k)

s(k) = trueWrite to ports

Write to portsD3 me(k)

1

1

D3 me(k)

Figure 5 Flow diagram of the receiver Arduino codes

The next step is the Delta modulation This kind ofmodulation can be viewed as an 1-bit ADC conversionscheme since it generates one output bit per input sampleThe scheme of the Delta modulation is shown in Figure 6 Itconsists of a comparator in the forward path and an integratorin the feedback path of a simple control loop The inputs ofthe comparator are the signal to be modulated119898(119896) and theoutput of the integrator 119909

119899(119896) As a result the modulated

output 119898119887(119896) is either true (high) or false (low) at any

given time as shown in Figure 7 In this figure we see aninput signal and the integral of the expression 119898(119896) minus 119909

119887(119896)

Clock

Delta modulatorTransmitter (Arduino)

Delta demodulatorReceiver (analog elect)

+

minus

int

intmb(k) mb(k) m(k)m(k)

xn(k)

Figure 6 Diagram of the simple Delta modulator

0 002 004 006 008 010246

Inpu

tIn signalRec signal

Time

(a)

0 002 004 006 008 01

012

Out

put

Time

minus1

(b)

Figure 7 Delta modulation example of a sine input signal (a)Input signal and reconstructed signal comparison (b) Modulatedoutput

For instance if 119898119887(119896) is ramping up and its output is less

than the input the integrator output will continue rampingup otherwise it will ramp down The signal 119898

119887(119896) is the

differential of the input and thus it can be reconstructed inthe receiver by integrating it In this work the integral signal119909119899(119896) is digitally generated by the Arduino program On

the other hand the reconstructing integrator of the receiveris implemented with analog electronics as will be explainedlater A full description of the Delta modulation techniquecan be found in Taylor [27]

After one bit from the Delta modulator is obtained thenext step is the message encryption In order to do so twologistic maps are called to generate two values 119909

1(119896) and

1199092(119896) The logistic maps have different initial conditions

that is 1199091(0) = 119909

2(0) Firstly the message is coded with

a value true or false that is assigned depending on thevalue 119909

1(119896) of the first chaotic map as can be seen in

Algorithm 1 (Part 1) where 119898119890(119896) is the encrypted message

and 119904(119896) is the key necessary to retrieve119898119890(119896)

In order to increase the security of the system the key119904(119896) is further encrypted following the same scheme It is

6 The Scientific World Journal

Part 1(1) if 119909

1(119896) gt 05 then

(2) 119898119890(119896) =119898

119887(119896)

(3) 119904(119896) = true(4) else(5) 119898

119890(119896) = 119898

119887(119896) ⊳Symbol means boolean negation

(6) 119904(119896) = false(7) end ifPart 2(8) if 119909

2(119896) lt 01 then

(9) 1199041(119896) = 119904(119896) ⊳Symbol means boolean negation

(10) 1199042(119896) = true

(11) else(12) 119904

1(119896) = 119904(119896)

(13) 1199042(119896) = false

(14) end ifPart 3(15) 119904119890(119896) = (119904

1(119896) AND 119904

2(119896)) OR (119904

1(119896) AND 119904

2(119896))

Algorithm 1

done by assigning it a value true or false that depends onthe second chaotic map value 119909

2(119896) as shown in Algorithm 1

(Part 2) where 1199041(119896) and 119904

2(119896) are auxiliary signals that are

used for encrypting and decrypting the key signal 119904(119896)The key is then finally encrypted by applying the

XOR function to the variables 1199041(119896) and 119904

2(119896) to yield

Algorithm 1 (part 3)The signals 119904

119890(119896)119898

119890(119896) and 119904

1(119896) are sent to the receiver

through digital outputs D2 D4 and D7 respectivelyIn the receiver the signals 119904

119890(119896) 119898

119890(119896) and 119904

1(119896) go

directly to the Arduino inputs D2 D4 and D7 respectivelyTheflowdiagramof the receiver program is shown in Figure 5as well The first step in decrypting the message is thedecryption of the key signal 119904

119890(119896) This is done by applying

the boolean formula that reverts the encryptionThe formulato calculate 119904

2(119896) given 119904

119890(119896) and 119904

1(119896) is obtained as follows

Recall that in the transmitter 119904119890(119896) is obtained by using the

XOR function

119904119890(119896) = 119904

1015840

1(119896) sdot 1199042(119896) + 119904

1(119896) sdot 119904

1015840

2(119896) (2)

where (sdot)1015840 is the complement operation of the correspondinglogic variable The truth table of the function in (2) is shownin Table 1 Thus given 119904

1(119896) and 119904

119890(119896) for obtaining the signal

1199042(119896) would result in the truth Table 2The Karnaugh maps technique [34] was used to find

the desired simplified expression for 1199042(119896) It is a pictorial

method in which the truth table of the boolean functionto be simplified is represented in a bidimensional formThe boolean variables are arranged according to the Graycode The terms of the simplified expression are found bygrouping 1s or 0s in an optimal way and therefore eliminatingunnecessary variables As a result the following booleanexpression for 119904

2(119896) is obtained

1199042(119896) = 119904

1015840

1(119896) sdot 119904119890(119896) + 119904

1(119896) sdot 119904

1015840

119890(119896) (3)

Table 1 Truth table for 119904119890(119896)

1199041(119896) 119904

2(119896) 119904

119890(119896)

0 0 00 1 11 0 11 1 0

Table 2 Truth table for 1199042(119896)

1199041(119896) 119904

119890(119896) 119904

2(119896)

0 0 00 1 11 0 01 1 1

Once 1199042(119896) is retrieved the signal 119904(119896) is obtained with

Algorithm 2 (part 1)The signal 119898

119890(119896) is finally decrypted by analyzing the

value of 119904(119896) (see Algorithm 2 (part 2)) where 119898119889(119896) is the

decrypted signal The output 119898119889(119896) is sent to the output pin

D3 and it goes directly to theDelta demodulator realizedwithanalog electronics using operational amplifiers

As shown in Figure 6 the Delta demodulation consists ofan integrator The signal is passed through different stages asshown in the circuit diagram of Figure 8The circuit has threemain blocksThe first one composed of the amplifiers U1 andU2 is a unipolar to bipolar converter Recall that the Arduinoinputsmust be unipolar so in the case that the original signalsare bipolar they must be recovered to its original form at theoutput of the Arduino Thus the signal 119898(119896) isin [0 5]V isconverted to a signal 119898(119905) isin [minus25 25]V The second blockis composed of amplifiers U3 and U4 They are designed tocompute the integral of the input signal It consists of anintegrator that performs the Delta demodulation (U4) and

The Scientific World Journal 7

1k

1k1k

m(t) me(k)

me(k)

s(k)

s(k)

mb(k)

mr(t)

+

+

+

+ +

+ +

+

+

+

+

minus

minus

minus

minus

minus minus

minus

TL082TL082

TL082

TL082 TL082

TL082 TL082

minus5V

minus5V minus5V

minus5V

minus5V minus5V

minus5V

minus5V

47120583F

47120583F

1k

1k

Vy

Vy

Vx

Vx

U6 U7

5V5V

5V 5V 5V

5V5V

U3

U2U1

U4 U5

01 120583F

10k 10k

10k

10k 10k

10k

10k

Arduino RX

Arduino TX

and+

and+

andminus

and+

andminus

andminus

and+

andminus

and+

andminus

and+

andminus

and+

andminus

22 k

33 120583F

Figure 8 Circuit diagram of the analog electronics in the receiver

an inverter amplifier (U3) to adjust the quality of its outputThese signals are finally sent through a low-pass filter anamplifier and an inverter (amplifiers U5ndashU7) to get the final119898119903(119905) which should be approximately equal to119898(119905)

42 Experimental Results For the experiments the logisticmaps were implemented with 119903 = 39018 and initial con-ditions 119909

1(0) = 01 and 119909

2(0) = 05 As an example the

sequence of numbers generated by the logistic map when119909(0) = 05 is shown in Figure 9 Each loop of the transmitteralgorithm is executed by the Arduino microcontroller in210 120583s approximately while each receiver loop is executedin 25 120583s approximately This means that the message signalbandwidth should be at most 500Hz approximately in orderto be well retrieved in the receiver

Figures 10 to 13 are screenshots of the oscilloscopecorresponding to the first experiment In this case a 125Hzsinewave 5 V peak-to-peak amplitude was used as amessagesignal In Figure 10 we see a comparison of the sent message119898(119905) (in blue) and the retrieved message 119898

119903(119905) (in yellow)

Figure 11 compares the sent message 119898(119905) (in blue) andthe encrypted message signal 119898

119890(119896) (in yellow) Figure 12

is a comparison on the sent message 119898(119905) (in blue) andthe encrypted key signal 119904

119890(119896) (in yellow) Finally Figure 13

compares the sent message (in blue) and the auxiliary signal1199041(119896) (in yellow)In subsequent experiments different frequencies and

waveforms were tested Figure 14 shows a 125Hz triangularwave message (in blue) and its retrieved version (in yellow)Figure 15 compares a 70Hz sine wave message (in blue) and

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

6 The Scientific World Journal

Part 1(1) if 119909

1(119896) gt 05 then

(2) 119898119890(119896) =119898

119887(119896)

(3) 119904(119896) = true(4) else(5) 119898

119890(119896) = 119898

119887(119896) ⊳Symbol means boolean negation

(6) 119904(119896) = false(7) end ifPart 2(8) if 119909

2(119896) lt 01 then

(9) 1199041(119896) = 119904(119896) ⊳Symbol means boolean negation

(10) 1199042(119896) = true

(11) else(12) 119904

1(119896) = 119904(119896)

(13) 1199042(119896) = false

(14) end ifPart 3(15) 119904119890(119896) = (119904

1(119896) AND 119904

2(119896)) OR (119904

1(119896) AND 119904

2(119896))

Algorithm 1

done by assigning it a value true or false that depends onthe second chaotic map value 119909

2(119896) as shown in Algorithm 1

(Part 2) where 1199041(119896) and 119904

2(119896) are auxiliary signals that are

used for encrypting and decrypting the key signal 119904(119896)The key is then finally encrypted by applying the

XOR function to the variables 1199041(119896) and 119904

2(119896) to yield

Algorithm 1 (part 3)The signals 119904

119890(119896)119898

119890(119896) and 119904

1(119896) are sent to the receiver

through digital outputs D2 D4 and D7 respectivelyIn the receiver the signals 119904

119890(119896) 119898

119890(119896) and 119904

1(119896) go

directly to the Arduino inputs D2 D4 and D7 respectivelyTheflowdiagramof the receiver program is shown in Figure 5as well The first step in decrypting the message is thedecryption of the key signal 119904

119890(119896) This is done by applying

the boolean formula that reverts the encryptionThe formulato calculate 119904

2(119896) given 119904

119890(119896) and 119904

1(119896) is obtained as follows

Recall that in the transmitter 119904119890(119896) is obtained by using the

XOR function

119904119890(119896) = 119904

1015840

1(119896) sdot 1199042(119896) + 119904

1(119896) sdot 119904

1015840

2(119896) (2)

where (sdot)1015840 is the complement operation of the correspondinglogic variable The truth table of the function in (2) is shownin Table 1 Thus given 119904

1(119896) and 119904

119890(119896) for obtaining the signal

1199042(119896) would result in the truth Table 2The Karnaugh maps technique [34] was used to find

the desired simplified expression for 1199042(119896) It is a pictorial

method in which the truth table of the boolean functionto be simplified is represented in a bidimensional formThe boolean variables are arranged according to the Graycode The terms of the simplified expression are found bygrouping 1s or 0s in an optimal way and therefore eliminatingunnecessary variables As a result the following booleanexpression for 119904

2(119896) is obtained

1199042(119896) = 119904

1015840

1(119896) sdot 119904119890(119896) + 119904

1(119896) sdot 119904

1015840

119890(119896) (3)

Table 1 Truth table for 119904119890(119896)

1199041(119896) 119904

2(119896) 119904

119890(119896)

0 0 00 1 11 0 11 1 0

Table 2 Truth table for 1199042(119896)

1199041(119896) 119904

119890(119896) 119904

2(119896)

0 0 00 1 11 0 01 1 1

Once 1199042(119896) is retrieved the signal 119904(119896) is obtained with

Algorithm 2 (part 1)The signal 119898

119890(119896) is finally decrypted by analyzing the

value of 119904(119896) (see Algorithm 2 (part 2)) where 119898119889(119896) is the

decrypted signal The output 119898119889(119896) is sent to the output pin

D3 and it goes directly to theDelta demodulator realizedwithanalog electronics using operational amplifiers

As shown in Figure 6 the Delta demodulation consists ofan integrator The signal is passed through different stages asshown in the circuit diagram of Figure 8The circuit has threemain blocksThe first one composed of the amplifiers U1 andU2 is a unipolar to bipolar converter Recall that the Arduinoinputsmust be unipolar so in the case that the original signalsare bipolar they must be recovered to its original form at theoutput of the Arduino Thus the signal 119898(119896) isin [0 5]V isconverted to a signal 119898(119905) isin [minus25 25]V The second blockis composed of amplifiers U3 and U4 They are designed tocompute the integral of the input signal It consists of anintegrator that performs the Delta demodulation (U4) and

The Scientific World Journal 7

1k

1k1k

m(t) me(k)

me(k)

s(k)

s(k)

mb(k)

mr(t)

+

+

+

+ +

+ +

+

+

+

+

minus

minus

minus

minus

minus minus

minus

TL082TL082

TL082

TL082 TL082

TL082 TL082

minus5V

minus5V minus5V

minus5V

minus5V minus5V

minus5V

minus5V

47120583F

47120583F

1k

1k

Vy

Vy

Vx

Vx

U6 U7

5V5V

5V 5V 5V

5V5V

U3

U2U1

U4 U5

01 120583F

10k 10k

10k

10k 10k

10k

10k

Arduino RX

Arduino TX

and+

and+

andminus

and+

andminus

andminus

and+

andminus

and+

andminus

and+

andminus

and+

andminus

22 k

33 120583F

Figure 8 Circuit diagram of the analog electronics in the receiver

an inverter amplifier (U3) to adjust the quality of its outputThese signals are finally sent through a low-pass filter anamplifier and an inverter (amplifiers U5ndashU7) to get the final119898119903(119905) which should be approximately equal to119898(119905)

42 Experimental Results For the experiments the logisticmaps were implemented with 119903 = 39018 and initial con-ditions 119909

1(0) = 01 and 119909

2(0) = 05 As an example the

sequence of numbers generated by the logistic map when119909(0) = 05 is shown in Figure 9 Each loop of the transmitteralgorithm is executed by the Arduino microcontroller in210 120583s approximately while each receiver loop is executedin 25 120583s approximately This means that the message signalbandwidth should be at most 500Hz approximately in orderto be well retrieved in the receiver

Figures 10 to 13 are screenshots of the oscilloscopecorresponding to the first experiment In this case a 125Hzsinewave 5 V peak-to-peak amplitude was used as amessagesignal In Figure 10 we see a comparison of the sent message119898(119905) (in blue) and the retrieved message 119898

119903(119905) (in yellow)

Figure 11 compares the sent message 119898(119905) (in blue) andthe encrypted message signal 119898

119890(119896) (in yellow) Figure 12

is a comparison on the sent message 119898(119905) (in blue) andthe encrypted key signal 119904

119890(119896) (in yellow) Finally Figure 13

compares the sent message (in blue) and the auxiliary signal1199041(119896) (in yellow)In subsequent experiments different frequencies and

waveforms were tested Figure 14 shows a 125Hz triangularwave message (in blue) and its retrieved version (in yellow)Figure 15 compares a 70Hz sine wave message (in blue) and

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

The Scientific World Journal 7

1k

1k1k

m(t) me(k)

me(k)

s(k)

s(k)

mb(k)

mr(t)

+

+

+

+ +

+ +

+

+

+

+

minus

minus

minus

minus

minus minus

minus

TL082TL082

TL082

TL082 TL082

TL082 TL082

minus5V

minus5V minus5V

minus5V

minus5V minus5V

minus5V

minus5V

47120583F

47120583F

1k

1k

Vy

Vy

Vx

Vx

U6 U7

5V5V

5V 5V 5V

5V5V

U3

U2U1

U4 U5

01 120583F

10k 10k

10k

10k 10k

10k

10k

Arduino RX

Arduino TX

and+

and+

andminus

and+

andminus

andminus

and+

andminus

and+

andminus

and+

andminus

and+

andminus

22 k

33 120583F

Figure 8 Circuit diagram of the analog electronics in the receiver

an inverter amplifier (U3) to adjust the quality of its outputThese signals are finally sent through a low-pass filter anamplifier and an inverter (amplifiers U5ndashU7) to get the final119898119903(119905) which should be approximately equal to119898(119905)

42 Experimental Results For the experiments the logisticmaps were implemented with 119903 = 39018 and initial con-ditions 119909

1(0) = 01 and 119909

2(0) = 05 As an example the

sequence of numbers generated by the logistic map when119909(0) = 05 is shown in Figure 9 Each loop of the transmitteralgorithm is executed by the Arduino microcontroller in210 120583s approximately while each receiver loop is executedin 25 120583s approximately This means that the message signalbandwidth should be at most 500Hz approximately in orderto be well retrieved in the receiver

Figures 10 to 13 are screenshots of the oscilloscopecorresponding to the first experiment In this case a 125Hzsinewave 5 V peak-to-peak amplitude was used as amessagesignal In Figure 10 we see a comparison of the sent message119898(119905) (in blue) and the retrieved message 119898

119903(119905) (in yellow)

Figure 11 compares the sent message 119898(119905) (in blue) andthe encrypted message signal 119898

119890(119896) (in yellow) Figure 12

is a comparison on the sent message 119898(119905) (in blue) andthe encrypted key signal 119904

119890(119896) (in yellow) Finally Figure 13

compares the sent message (in blue) and the auxiliary signal1199041(119896) (in yellow)In subsequent experiments different frequencies and

waveforms were tested Figure 14 shows a 125Hz triangularwave message (in blue) and its retrieved version (in yellow)Figure 15 compares a 70Hz sine wave message (in blue) and

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

8 The Scientific World Journal

Part 1(1) 1199042(119896) = (119904

1(119896) AND 119904

119890(119896)) OR (119904

1(119896) AND 119904

119890(119896))

(2) if 1199042(119896) = true then

(3) 119904(119896) = 1199041(119896)

(4) else(5) 119904(119896) = 119904

1(119896)

(6) end ifPart 2(7) if 119904(119896) = true then(8) write to port D3119898

119889(119896) = 119898

119890(119896)

(9) else(10) write to port D3119898

119889(119896) = 119898

119890(119896)

(11) end if

Algorithm 2

0 50 100 1500

02

04

06

08

1

x[i]

i

Figure 9 Numbers generated by the logistic map with 119903 = 39018and 119909(0) = 05

Figure 10 125Hz sine wave message Blue sent message Yellowretrieved message

its retrieved version (in yellow) Finally Figure 16 shows arandom wave message (in blue) and its retrieved version (inyellow)This signal was generated bymaking sounds throughan electret microphone

5 Conclusion

In this paper we presented a communication system basedon chaotic logistic maps and an experimental realization ofit The proposed communication system uses a simple Deltamodulator to modulate the message signal and a logistic

Figure 11 125Hz sine wave message Blue sent message Yellowencrypted message signal

Figure 12 125Hz sine wave message Blue sent message Yellowencrypted key

Figure 13 125Hz sine wave message Blue sent message Yellowauxiliary signal 119904

1(119896)

map for encryption A key signal is also generated andencrypted in order to retrieve the message in the receiverside without the need for synchronization The whole systemwas implemented with Arduino Uno microcontroller boardsthat run the encryption and decryption algorithms in thetransmitter and receiver respectively The experiment resultsshowed the feasibility of using the Arduino microprocessorsfor the task proposed With the proposed scheme it ispossible to transmit signals whose bandwidth is 500Hzapproximately

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

The Scientific World Journal 9

Figure 14 125Hz triangular wave message Blue sent messageYellow retrieved message

Figure 15 70Hz sine wave message Blue sent message Yellowretrieved message

Figure 16 Random wave message Blue sent message Yellowretrieved message

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

First author is supported by the fellowship fromCAPESPrograma Nacional de Pos-Doutorado from BrazilThis work was funded by the European Union (EuropeanRegional Development Fund) and the Spanish Ministryof Economy and Competitiveness through the research

projects DPI2012-32375FEDER DPI2011-28033-C03-01 andDPI2014-58427-C2-1-R and by the Government of Catalonia(Spain) through 2014SGR859

References

[1] L Larger and J-P Goedgebuer ldquoEncryption using chaoticdynamics for optical telecommunicationsrdquo Comptes RendusPhysique vol 5 no 6 pp 609ndash611 2004

[2] C K Volos ldquoChaotic random bit generator realized with amocrocontrollerrdquo Journal of Computations amp Modelling vol 3no 4 pp 115ndash136 2013

[3] C-K Chen and C-L Lin ldquoText encryption using ECG signalswith chaotic Logistic maprdquo in Proceedings of the 5th IEEEConference on Industrial Electronics and Applications (ICIEArsquo10) pp 1741ndash1746 Taichung Taiwan June 2010

[4] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bifur-cation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[7] M Zapateiro Y Vidal and L Acho ldquoA secure communicationscheme based on chaotic Duffing oscillators and frequencyestimation for the transmission of binary-coded messagesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 4 pp 991ndash1003 2014

[8] M Zapateiro De la Hoz L Acho and Y Vidal ldquoA modifiedChua chaotic oscillator and its application to secure commu-nicationsrdquo Applied Mathematics and Computation vol 247 pp712ndash722 2014

[9] S Hammami ldquoState feedback-based secure image cryptosystemusing hyperchaotic synchronizationrdquo ISA Transactions vol 54pp 52ndash59 2015

[10] K Fallahi and H Leung ldquoA chaos secure communicationscheme based on multiplication modulationrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 15 no 2 pp368ndash383 2010

[11] S T Liu and F Y Sun ldquoSpatial chaos-based image encryptiondesignrdquo Science in China Series G Physics Mechanics andAstronomy vol 52 no 2 pp 177ndash183 2009

[12] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[13] C Li S Li M Asim J Nunez G Alvarez and G Chen ldquoOnthe security defects of an image encryption schemerdquo Image andVision Computing vol 27 no 9 pp 1371ndash1381 2009

[14] P-H Lee S-C Pei and Y-Y Chen ldquoGenerating chaotic streamciphers using chaotic systemsrdquo Chinese Journal of Physics vol41 no 6 pp 559ndash581 2003

[15] S Shyamsunder and G Kaliyaperumal ldquoImage encryption anddecryption using chaotic maps and modular arithmeticrdquo TheAmerican Journal of Signal Processing vol 1 no 1 pp 24ndash332011

[16] L Y Zhang X Hu Y Liu K-W Wong and J Gan ldquoA chaoticimage encryption scheme owning temp-value feedbackrdquo Com-munications inNonlinear Science andNumerical Simulation vol19 no 10 pp 3653ndash3659 2014

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953

10 The Scientific World Journal

[17] C K Volos I M Kyprianidis and I N Stouboulos ldquoTheeffect of foreign direct investment in economic growth fromthe perspective of nonlinear dynamicsrdquo Journal of EngineeringScience and Technology Review vol 8 no 1 pp 1ndash7 2015

[18] J Miskiewicz and M Ausloos ldquoA logistic map approach toeconomic cycles (I) The best adapted companiesrdquo Physica AStatistical Mechanics and its Applications vol 336 no 1-2 pp206ndash214 2004

[19] D K Campbell and G Mayer-Krees ldquoChaos and politicsapplications of nonlinear dynamics to socio-political issuesrdquo inThe Impact of Chaos on Science and Society pp 18ndash63 UnitedNations University Press 1997

[20] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[21] M A Murillo-Escobar F Abundiz-Perez C Cruz-Hernandezand RM Lopez-Gutierrez ldquoA novel symmetric text encryptionalgorithm based on logistic maprdquo in Proceedings of the Inter-national Conference on Communications Signal Processing andComputers (ICNC rsquo14) Honolulu Hawaii USA February 2014

[22] C K Volos N Doukas I M Kyprianidis I N Stouboulos andT G Kostis ldquoChaotic autonomous mobile robot for militarymissionsrdquo in Proceedings of the 17th International Conference onCommunications Rhodes Island Greece July 2013

[23] A Pande and J Zambreno ldquoA chaotic encryption schemefor real-time embedded systems design and implementationrdquoTelecommunication Systems vol 52 no 2 pp 551ndash561 2013

[24] A J Lawrance and R C Wolff ldquoBinary time series generatedby chaotic logistic mapsrdquo Stochastics and Dynamics vol 3 no4 pp 529ndash544 2003

[25] S-M Chang ldquoChaotic generator in digital secure communica-tionrdquo in Proceedings of theWorld Congress on Engineering (WCErsquo09) London UK July 2009

[26] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[27] D S Taylor ldquoDesign of continuously variable slope deltamodu-lation communication systemsrdquoMotorola Technical DocumentAN1544 1996

[28] Arduino httpstorearduinoccproductA000066[29] J Kint D Constales and A Vanderbauwhede ldquoPierre-Francois

Verhulstrsquos final triumphrdquo in The Logistic Map and the Route toChaos M Ausloos and M Dirickx Eds pp 13ndash28 SpringerHeidelberg Germany 2006

[30] H Pastijn ldquoThe logistic map and the route to chaosrdquo in ChaoticGrowth with the Logistic Model of P-F Verhulst M Ausloos andM Dirickx Eds p 3 Springer Heidelberg Germany 2006

[31] P F Verhulst ldquoRecherches mathematiques sur la loi drsquoaccroiss-ement de la populationrdquo Memoires de lrsquoAcademie Royale desSciences des Lettres et des Beaux-Arts de Belgique vol 18 pp1ndash38 1845

[32] P F Verhulst ldquoDeuxieme memoire sur la loi drsquoaccroissement dela populationrdquo Memoires de lrsquoAcademie Royale des Sciences desLettres et des Beaux-Arts de Belgique vol 20 pp 1ndash32 1847

[33] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[34] M Karnaugh ldquoThemapmethod for synthesis of combinationallogic circuitsrdquoTransactions of the American Institute of ElectricalEngineers Part I Communications and Electronics vol 72 pp593ndash599 1953