CityU-PolyU Joint Seminar Series on Chaos and System ... · Future Work Page 2 of 53 JJ II J I ←-...

CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

of 53

JJ II

J I

←↩ ↪→Full Screen

Search

Close

Chaos SeminarCD634, PolyU7 May 2004

– CityU-PolyU Joint Seminar Series on Chaos and System Control –

On Digital Degradation ofChaos-Based Analog Device

Implementation

Shujun Li

Center for Chaos Control and Synchronization,Department of Electronic Engineering, CityU HK

[email protected]://www.hooklee.com

http://www.ee.cityu.edu.hk/%7Ecccs/main.htm

mailto:[email protected]

http://www.hooklee.com

CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

of 53

JJ II

J I


Search

Close


Before Starting

� The Name of The Game

– digital chaotic systems (or digital chaos)

= computer (or computerized) ∼= discrete-time and discrete-value ∼ [W.Schwarz et al.]

= 2−n-discretized ∼ [M. Blank]

= finite-state ∼ [Masuda & Aihara]

= discrete chaos [Waelbroeck & Zertuche]

= pseudo chaos [Chirkikov & Vivaldi]

– digital chaotic orbits = pseudo orbits

� Motivation of the Research


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

a of 53

JJ II

J I


Search

Close


Outline of This Talk

� Related Work

� Preliminaries

� Major Results

� Applications

– Theoretical and experimental observations:there really exists non-trivial dynamical degra-dation in digital chaotic systems.

– Practical countermeasures: how to over-come the dynamical degradation of digitalchaotic systems in practice?


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 3b of 53

JJ II

J I


Search

Close



� Related Work

� Preliminaries

� Major Results

� Applications

– Piecewise linear chaotic maps (PWLCM)and their dynamical properties

– Digital normalization of the unit interval inn-bit fixed-point arithmetic


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 3c of 53

JJ II

J I


Search

Close



� Related Work

� Preliminaries

� Major Results

� Applications

A new series of dynamical indicators are proposedto quantitatively measure the degradation level ofdigital PWLCM implemented under fixed-point fi-nite precision.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 3d of 53

JJ II

J I


Search

Close



� Related Work

� Preliminaries

� Major Results

� Applications

– Performance comparison of different meth-ods for improving dynamical degradation ofdigital chaotic systems

– Chaotic cryptography

– Chaotic pseudo-random number generation

⇓Some common principles for implementation


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 4 of 53

JJ II

J I


Search

Close


1 Related Work

Two Types of Digital Chaotic Systems

– Type-I: the continuous chaotic systems is numericallyrealized in digital computers, under floating-point finiteprecision or fixed-point finite precision.

Given a 1-D discrete-time chaotic map F : X → X, theType-I digital map F ′ is the composition of F and a quan-tization function G : X′→X′, where X′ is the finite versionof the real interval X, shown as

F ′ = G◦F : X′→ X′.

Three common quantization functions used in digitalworld: round, truncation/floor, and ceiling functions.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 5a of 53

JJ II

J I


Search

Close



– Type-II: the continuous chaotic equation is re-defined(i.e., discretized) in digital forms, or the equation itself isoriginally defined in digital forms.

Example 1 1-D digital Logistic map defined over8-bit digital space {0, · · · ,255} [Kocarev & Jaki-moski, PLA/IEEETCASI 2001]: given x̃(k + 1) =bx(k) · (256−x(k))/64c,

x(k+1) =

{x̃(k+1), 0≤ x̃(k+1)≤ 255,

255, x̃(k+1) = 256.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 5b of 53

JJ II

J I


Search

Close




Example 2 1-D digital skew tent map defined overthe digital space {1, · · · ,M} [Masuda & Aihara,IEEETCASI/IJBC 2001]: for 1≤ A≤M,

x(k+1)=

{⌈MA ·x(k)

⌉, 1≤ x(k)≤ A,⌊

MM−A · (M−x(k))

⌋+1, A < x(k)≤M.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 5c of 53

JJ II

J I


Search

Close




Example 3 2-D digital Baker map defined over N×N lattice [Fridrich, IJBC 1998]: given n1+ · · ·+nk =n and Ni = n1+ · · ·+ni , for Ni ≤ x(k) < Ni +ni+1 and0≤ y(k) < N,{

x(k+1) = Nni

(x(k)−Ni)+y(k) mod Nni

,

y(k+1) = niN (y(k)−y(k) mod N

ni)+Ni .


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 6 of 53

JJ II

J I


Search

Close



– Type-I: the continuous chaotic systems is numer-ically realized in digital computers, under floating-point finite precision or fixed-point finite precision.

∈– Type-II: the continuous chaotic equation is re-defined (i.e., discretized) in digital forms, or theequation itself is originally defined in digital forms


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 7a of 53

JJ II

J I


Search

Close


Pathologies of Digital Chaotic Systems

– Intractable Quantization Errors: the errors are intro-duced in a very complex and uncontrollable (chaotic?)way. As a result, the chaotic orbits will dramaticallychange, even their topological structures may change.

Random Perturbation Theory is not so good.

Example 1 (Fryska & Zohdy 1992) Continuous orbits(single-scroll) 6= digital orbits in single-precision (32-bit)floating-point arithmetic (two-scroll) 6= digital orbits in ex-tended double-precision (80-bit) floating-point arithmetic(periodic, no scroll).


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 7b of 53

JJ II

J I


Search

Close




Example 2 (Palmore & Herring 1990) The change ofarithmetical details will definitely change the structure ofthe digital chaotic orbits.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 7c of 53

JJ II

J I


Search

Close




Example 3 The extreme example on tent mapF(x) = 1−2|x−0.5|: invoke matlab to show the con-vergence of all digital orbits to zero within about 54iterations.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 7d of 53

JJ II

J I


Search

Close




Note: following IEEE 754 standard, a 64-bit floating-pointnumber b63b62 · · ·b52b51 · · ·b0 represents the following bi-nary real number (except for some special values)

called mantissa︷︸︸︷

±1.b51 · · ·b0︸︷︷︸

53 bits

×2b62 · · ·b52−1023.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 8 of 53

JJ II

J I


Search

Close



– Long-Term Dynamics – Unavoidable Periodicity: itis obvious that all digital chaotic orbits will be eventuallyperiodic.

x0 xl

xl+n

xl+1

x1 ...... ......

transient length = l

cycle period = nA typical digital chaotic orbit, where both n = 0 and

l = 0 are possible.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 9 of 53

JJ II

J I


Search

Close


Example 1 The digital skew tent map

F(x) =

{x/p, 0≤ x≤ p

(1−x)/(1− p), p≤ x≤ 1

with the control parameter p = 3/24 = 0.1875, realized in4-bit fixed-point arithmetic and the round-off quantization

function. Here, node i denotes x =i

24 .

0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 10 of 53

JJ II

J I


Search

Close



– Long-Term Dynamics – Unavoidable Periodicity: ex-tensive research has revealed a scaling-law on the period-icity of digital chaotic systems.

* Assume the cardinality of the finite set is N.

� The mean transient lengths, cycle periods of all dig-ital orbits yield O(Nd), where d∈ (0,1) and is deter-mined by the chaotic equation; generally Nd� N.

� The number of attractive cycles (including fixedpoints) yields O(logN).

� The distribution of cycle period is non-uniform orGaussian-like, but (roughly) monotonic decreasing,which means there are a large number of pseudoorbits with short cycle period.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 11 of 53

JJ II

J I


Search

Close



– Long-Term Dynamics – Unavoidable Periodicity

� The number of attractive cycles (including fixedpoints) yields O(logN).

0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 12 of 53

JJ II

J I


Search

Close



– Incapability of the shadowing lemma: the followingfacts make this lemma problematic:

� The topological structure of the orbits may be com-pletely changed;

� No infinite orbits occur in computers;

� The stability of digital chaotic orbits may be differ-ent from the shadowing ones;

� All shadowing orbits may (or always?) be of measurezero.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 13 of 53

JJ II

J I


Search

Close


More Pathologies?

� The ergodicity of continuous chaotic systems be-comes meaningless in digital world, since all digi-tal chaotic states are of measure zero in continuousphase space.

� To measure the degradation of digital chaotic sys-tems, the dynamical properties should be redefinedin digital forms.

� There is not yet an established theory on the studyof digital chaotic systems, so it’s hard to do more.

M. Blank, Discreteness and Continuity in Problemsof Chaotic Dynamics, vol. 161 of Translation ofMathematical Monographs, American MathematicalSociety, Providence, Rhode Island, 1997


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 14 of 53

JJ II

J I


Search

Close


Practical Countermeasures

� Using higher finite precision;

� Cascading multiple chaotic systems;

� Pseudo-randomly perturbing the chaotic orbit

and/or the control parameter.

Based on our results on digital PWLCM, the performanceof the above three methods will be compared.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 15 of 53

JJ II

J I


Search

Close


2 Preliminaries

Piecewise Linear Chaotic Map (PWLCM)

Given a real interval X = [α,β] ⊂ R, a PWLCM issuch a map F : X→ X:

i = 1∼m,F(x)|Ci = Fi(x) = aix+bi ,

where {Ci}mi=1 is a partition of X.

� If ∀i = 1∼m,Fi(Ci) = X, we say the above PWLCMsatisfies the onto property.

� Any 1D PWLM is topological conjugate to its nor-malized version F[0,1] defined on the unit intervalI = [0,1] via an affine transform.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 16a of 53

JJ II

J I


Search

Close


Two Examples of PWLCM with the OntoProperty

0 p=0.3 0.5 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Skew Tent map with a control parameter p = 0.3C1 = [0, p), C2 = [p,1]


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 16b of 53

JJ II

J I


Search

Close


Two Examples of PWLCM with the OntoProperty

0 p=0.3 0.5 1−p=0.7 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1∑m

i=1 sign(ai) · ‖Ci‖2 = 0

Zhou’s PWLCM with a control parameter p = 0.3 [Zhou& Ling, IEEETCASI 1997]

C1 = [0, p), C2 = [p,1/2], C3 = [1/2,1− p), C4 = [1− p,1]


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 17 of 53

JJ II

J I


Search

Close


PWLCM with the Onto Property

1. Its Lyapunov exponent is positive.

2. It is ergodic, mixing and exact.

3. It has a uniform invariant density f (x) = 1β−α ;

4. Its auto-correlation function

τ(n) =1

σ2 limN→∞

1N

N−1

∑i=0

(xi− x̄)(xi+n− x̄)→ 0

as n→ ∞.

Especially, if ∑mi=1 sign(ai) · ‖Ci‖2 = 0 is satisfied,

τ(n) = δ(n). This conditions actually meas the sumof the slopes of m linear segments are zero. Zhou’smap is a case.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 18 of 53

JJ II

J I


Search

Close


PWLCM with the Onto Property

In continuous world, the uniform invariant densityfunction of onto PWLCM means that uniform in-put will generate uniform output, and that thechaotic orbit from almost every initial condi-tion will lead to the same uniform distributionf (x) = 1/(β−α).

6⇓

In digital world, for a onto PWLCM, discrete uni-form input can not generate discrete uniformoutput, or a uniform random variable will be-come non-uniform after digital chaotic itera-tions.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 19 of 53

JJ II

J I


Search

Close


The Focus of Our Work

A new question: can we exactly measure the non-uniformity of the chaotic output of a digital 1DPWLCM with a (discrete) uniform input?

A new arithmetic viewpoint: to study digital chaoticsystems by quantitatively investigating how thechaotic iterations are calculated in computers.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 20 of 53

JJ II

J I


Search

Close


The Target of Our Work

The normalized PWLCM F[0,1] realized with thefixed-point arithmetic.

* Why fixed-point?

� Floating-point arithmetic is much more com-plicated than fixed-point one, due to the non-uniformity of floating-point numbers in the wholedigital space.

� Floating-point calculation units are much slowerand much expensive than fixed-point one, so manychaos-based digital devices employ fixed-point arith-metic to achieve a fast speed and save the imple-mentation cost.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 21a of 53

JJ II

J I


Search

Close


An Illustrative Example

Digital tent map F(x) = 1− 2|x− 0.5| realized in 4-bitfixed-point arithmetic

0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9

x distributes uniformly: Prob[x = i/24] = 1/17


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 21b of 53

JJ II

J I


Search

Close




0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9

P3=0/17

P8=P13=0/17

P0=2/17P5=2/17

P11=2/17

the distribution of F(x) becomes non-uniform


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 21c of 53

JJ II

J I


Search

Close




0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9

P3=0/17

P8=P13=0/17

P0=3/17P5=2/17

P11=2/17

P16=0/17 P6=2/17

P14=2/17P4=0/17

the distribution of F2(x) becomes even more non-uniform


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 21d of 53

JJ II

J I


Search

Close




0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9

P0=3/17 P12,5,14,2,11,6=13/17

P9=1/17

k = 4∼ ∞: the distribution of Fk(x)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 22 of 53

JJ II

J I


Search

Close


Notations and Concepts of the DigitizedUnit Interval in Fixed-Point Arithmetic

Definition 1 A discrete set Sn = {a|a = ∑ni=1ai ·2−i ,ai ∈

{0,1}} is called a digital set with resolution (or pre-cision) n. ∀i < j, Si is called the digital subset of Sj

with resolution i. Specially, define S0 = {0},S∞ = [0,1],we have {0}= S0⊂ S1⊂ ·· · ⊂ S∞ = [0,1].

Definition 2 Define Vi = Si−Si−1 (i ≥ 1) and V0 = S0. Vi

is called a digital layer with resolution i. ∀p ∈ Vi , i is

called the resolution of p .

The resolution of a binary decimal p∈ Vi is actually theposition of its least 1-bit in the binary representation, i.e.,Vi = {a|a∈ Si and ai = 1}.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 23 of 53

JJ II

J I


Search

Close


Notations and Concepts of the DigitizedUnit Interval in Fixed-Point Arithmetic

Definition 3 A function Gn : S∞ = [0,1]→ Sn is called adigital approximate transformation function (DATF)with resolution n, if ∀x∈ [0,1], |Gn(x)−x|< 1/2n.

Three most frequently-used DATF in digital arithmetic are

1. roundnnn(((xxx))) = round(x ·2n)/2n;

2. floornnn(((xxx))) = bx ·2nc/2n;

3. ceilnnn(((xxx))) = dx ·2ne/2n.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 24 of 53

JJ II

J I


Search

Close


3 Major Results

� The Representation of Digital PWLCM

� The Proposed Dynamical Indicators

� Two Typical Examples: Zhou’s PWLCM and SkewTent Map

� Generic PWLCM with/without the Onto Property

� The Condition of Fk(x)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 25 of 53

JJ II

J I


Search

Close


The Representation of Digital PWLCM

When a normalized 1-D PWLCM F : S∞ = [0,1]→ S∞ =[0,1] is realized in the digital set Sn, it will become adigital composition map F ′n = Gn◦F : S′n→S′n, where S′n =Sn∪{1}.With a saturation function fs(x) : [0,1]→ [0,1)

fs(x) =

{0, x = 1,

x, 0≤ x < 1,

we study the following digital map well-defined over Sn:

Fn = fs◦F ′n = fs◦Gn◦F : Sn→ Sn.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 26 of 53

JJ II

J I


Search

Close


The Proposed Dynamical Indicators

When x satisfies a discrete uniform distribution over Sn,

� n resolution-dependent degradation factors: ∀ j =1∼ n, the probability of the least j bits of Fn(x)

are all zeros, i.e., Pj = Prob[Fn(x) ∈ Sn− j ].

� an average degradation factor P =1n

n∑j=1

Pj

2− j .

The computability of the n+1 indicators is solved,and the relationship between P1∼Pn, P and the con-trol parameter(s) of PWLCM is clarified.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 27 of 53

JJ II

J I


Search

Close


A Typical Example: Zhou’s PWLCM

0 p 0.5 1−p 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Zhou’s PWLCM with a control parameter p∈ (0,0.5)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 28 of 53

JJ II

J I


Search

Close



Theorem 1 Assume that a discrete random variable x dis-tributes uniformly in Sn. For digital Zhou’s PWLCM, if p’sresolution is i (2≤ i ≤ n), the following results are true:

1. When Gn(·) = roundn(·),

Pj =

4/2 j , i ≤ j ≤ n,

4/2i , j = i−1,

1/2 j , 1≤ j ≤ i−2;

when Gn(·) = floorn(·) or ceiln(·),

Pj =

{4/2 j , i ≤ j ≤ n,

1/2 j +2/2i , 1≤ j ≤ i−1.

2. ∀k∈ {0, · · · ,2n−i−1},

Prob[floorn−i(Fn(x, p)) = k/2n−i] = 1/2n−i .


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 29a of 53

JJ II

J I


Search

Close



1 2 3 4 5 6 7 8 9 10−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

j

log 2

Pj

log2Pj (1≤ j ≤ n) of p = 3/16∈V4, where n = 10 andGn(·) = roundn(·) (better)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 29b of 53

JJ II

J I


Search

Close



1 2 3 4 5 6 7 8 9 10−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

j

log 2

Pj

log2Pj (1≤ j ≤ n) of p = 3/16∈V4, where n = 10 andGn(·) = floorn(·) or ceiln(·) (worse)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 30 of 53

JJ II

J I


Search

Close



Theorem 2 Assume that a discrete random variable x dis-tributes uniformly in Sn. For digital Zhou’s PWLCM, thevalue of a degradation factor Pi (i = 1∼ n) can be deter-mined from the resolution of p as follows:

1. if p’s resolution is from 2 to i, Pi = 4/2i ;2. if p’s resolution is i +1, Pi = 2/2i ;3. if p’s resolution is larger than i +1,

Pi =

{1/2i , Gn(·) = roundn(·),1/2i +2/2 j , Gn(·) = floorn(·) or ceiln(·).

This theorem shows that the resolution of the con-trol parameter p can be distinguished from the val-ues of P1∼ Pn.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 31a of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

p

log 2

P 1

log2P1 vs. p, where n = 10 and Gn(·) = floorn(·)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 31b of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

p

log 2

P 2



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 31c of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

p

log 2

P 3



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 31d of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−4

−3.8

−3.6

−3.4

−3.2

−3

−2.8

−2.6

−2.4

−2.2

−2

p

log 2

P 4



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 31e of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

−3

p

log 2

P 5



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 31f of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6

−5.8

−5.6

−5.4

−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

p

log 2

P 6



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 32 of 53

JJ II

J I


Search

Close



Definition 4 For PWLCM with two control parametersp1 6= p2, we say p1 is weaker than p2 if P(p1) > P(p2)and denote their relation by ppp111≺≺≺ ppp222.

Corollary 1 For the digital Zhou’s PWLCM, given twodifferent control parameters p1 ∈ Vi1, p2 ∈ Vi2, it is true

that p1≺ p2⇔ i1 < i2 .

This corollary tells us that the non-uniformity ofchaotic output can be quantitatively measured withthe resolution of the control parameter of theconcerned digital PWLCM.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 33 of 53

JJ II

J I


Search

Close



2 3 4 5 6 7 8 9 100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

the resolution i

log 2

P

log2P vs. i (the resolution of p)◦ - roundn(·) (better), ♦ - floorn(·) or ceiln(·) (worse)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 32 of 53

JJ II

J I


Search

Close


Another Typical Example: Skew Tent Map

0 p 0.5 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Skew Tent map with a control parameter p∈ (0,1)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 33 of 53

JJ II

J I


Search

Close



Theorem 3 (vs. Theorem 1) Assume that a discreterandom variable x distributes uniformly in Sn. For digi-tal skew tent map, if p’s resolution is i (1≤ i ≤ n), thefollowing results are true:

1. When Gn(·) = roundn(·),

Pj =

{2/2 j , i ≤ j ≤ n,

1/2 j−1, 1≤ j ≤ i−1;

when Gn(·) = floorn(·) or ceiln(·),

Pj =

{2/2 j , i ≤ j ≤ n,

1/2 j +1/2i , 1≤ j ≤ i−1.

2. ∀k ∈ [0,2n−i − 1], P{floorn−i(Fn(x, p)) = k/2n−i} =1/2n−i .


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 34 of 53

JJ II

J I


Search

Close



Theorem 4 (vs. Theorem 2) Assume that a discreterandom variable x distributes uniformly in Sn. For thedigital skew tent map, the value of a degradation factorPi (i = 1∼ n) can be determined from the resolution of pas follows:

1. if p’s resolution is from 1 to i, Pi = 2/2i ;

2. if p’s resolution is larger than i,

Pi =

{1/2i , Gn(·) = roundn(·),1/2i +1/2 j , Gn(·) = floorn(·) or ceiln(·).

Corollary 2 (vs. Corollary 1) For digital skew tent map,given two different control parameters p1 ∈Vi1, p2 ∈Vi2, itis true that p1≺ p2⇔ i1 < i2.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 35 of 53

JJ II

J I


Search

Close


Generic PWLCM with the Onto Property

Computability of the Dynamical Indicators: calculatingthe values of Pj on all linear segments and then accumu-lating them.

Fi(x) = aix+ bi ⇒ Fi(x′i) = x′i/pi , x′i ∈ [0, pi), where pi =1/|ai | and x′i = sign(ai) · (x+bi/ai).

0 1 0

1

pi


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 36 of 53

JJ II

J I


Search

Close



Theorem 5 Assume that a discrete random variable x dis-tributes uniformly in the discrete set [0, p)∩Sn. For a digi-tal linear function Fn(x) = Gn(x/p), if p= Np/2i ∈Vi (1≤i ≤ n), where Np is an odd integer in {1, · · · ,2i−1},

� when 1≤ j ≤ i−1,

Pj =

bNp/2 jc+1

Np, Gn(·) = floorn(·) or ceiln(·),

2·bNp/2 j+1c+1Np

, Gn(·) = roundn(·).

� when i ≤ j ≤ n, Pj = 1Np·2 j−i ;

It’s possible to calculate the dynamical indicators onall linear segments and combine them.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 37 of 53

JJ II

J I


Search

Close



Assume the resolution of each pi is r i , from Theorem 5,we can deduce

Pj = ∑mi=1P(i)

j .

Here, P(i)j is calculated as follows:

� when r i ≤ j ≤ n, P(i)j = 1/2 j ;

� when 1≤ j ≤ r i−1,

P(i)j =

bNpi /2 jc+1

2ri , Gn(·) = floorn(·) or ceiln(·),2·bNpi /2 j+1c+1

2ri , Gn(·) = roundn(·).


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 38 of 53

JJ II

J I


Search

Close



How Does PPP jjj Changes as jjj and the Resolutions?

� j = n→maxmi=1(r i):

Pj =m2 j ⇒

Pj

2− j = m;

� j = (minmi=1(r i)−1)→ 1:∣∣∣∣Pj −

12 j

∣∣∣∣ <m

∑i=1

12r i→

Pj

2− j ≈ 1

� j =(maxm

i=1(r i)−1)→minm

i=1(r i):

Pj

2− j = m 1.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 39 of 53

JJ II

J I


Search

Close




1 3 5 6 7 8 9 101

2

3

5

m = 4

(maxmi=1 ri)(minm

i=1 ri)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 40 of 53

JJ II

J I


Search

Close




The smaller the resolutions of the slopes of all linear

segments, the larger P = 1n ∑m

j=1Pj

2− j and the worsethe dynamical degradation of the PWLCM.

Q: Why does a smaller resolution of the linear slope leadto a larger dynamical degradation?

A: Rewrite p as p =Np

2i = 2n−i ·Np

2n , we can see a smaller

resolution i actually means a larger multiplication factor2n−i .


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 41 of 53

JJ II

J I


Search

Close


More Generic PWLCM without the OntoProperty

The condition will become much more complicated thanonto cases, but theoretical analysis shows that the dynam-ical indicators are still computable.

0 1 0

1

pipi,1 pi,2

The basic idea of the computability


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 42 of 53

JJ II

J I


Search

Close


When Chaotic Iterations Go On: Fk(x)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

−3

p

log 2

P 5



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 43 of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

p

log 2

P 5

log2P5 of F2n (x) vs. p


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 44 of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6

−5

−4

−3

−2

−1

0

p

log 2

P 5



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 45 of 53

JJ II

J I


Search

Close



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−7

−6

−5

−4

−3

−2

−1

0

p

log 2

P 5



CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 46 of 53

JJ II

J I


Search

Close



2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

the resolution i

log 2

P

k = 1→ 32

log2P of Fk(x) (k = 1∼ 32) vs. i, where Gn(·) = floorn(·)


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 47 of 53

JJ II

J I


Search

Close



0

16

3

1

13

15

1412 45

7 10

6

11

2 8

9

P0=3/17 P12,5,14,2,11,6=13/17

P9=1/17


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 48a of 53

JJ II

J I


Search

Close


Applications

I: Comparing the Practical Methods of Im-proving the Dynamical Degradation

� Using higher finite precision

� Cascading multiple chaotic systems

� Pseudo-randomly perturbing the control parameter


– can prolong the mean length of all chaotic orbits;

– cannot improve the weakness level of all control param-eters in low precisions.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 48b of 53

JJ II

J I


Search

Close


Applications







– cannot improve the non-uniformity at all.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 48c of 53

JJ II

J I


Search

Close


Applications







– can make weak control parameters stronger, but alsomake strongest ones a little weaker.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 48d of 53

JJ II

J I


Search

Close


Applications







– can make all control parameters sufficiently strong;

– the best method of improving the degradation.

– can be combined with the perturbation on the controlparameter to achieve an even better performance.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 49 of 53

JJ II

J I


Search

Close


II: Chaotic Cryptography

Use the deterministic relationship between the dy-namical indicators and the control parameters ofPWLCM to find weak keys or security defects.

Perturbing signal

m-sequence

Zhou’sPWLCM

K

⊕

Plaintext

Ciphertext

[IEEETCASI 1997]

[AES 1998]Initialcondition )(0 tu

⊕)(1 tu )('1 tu

Zhou’sPWLCM

K

)('1 tuk− ( )ku t⊕ ⊕ '( )ku t

Zhou’s chaotic ciphers

Example 1 By removing the last round perturbation,uk(t) approximately satisfies the theorems and the corol-lary of digital Zhou’s PWLCM. Then, based on the deter-mined resolutions, the key space can be decomposedinto nnn sub-spaces with different weak levels.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 50 of 53

JJ II

J I


Search

Close


II: Chaotic Cryptography

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

p

log 2

P 5

log2P5 of uk(t) vs. p


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 51 of 53

JJ II

J I


Search

Close


III: Chaotic Pseudo-Random Number Gen-erators

The chaotic orbit of digital PWLCM will neverbe discretely uniform. Some extra postprocessingshould be added to improve the performance ofPRNG based digital PWLCM.

* Some suggested common principles:

� use the strongest control parameters, i.e., those withthe largest resolution;

� realize digital chaotic systems with pseudo-randomperturbation;

� use round quantization function instead of floor andceiling functions.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 52 of 53

JJ II

J I


Search

Close


4 Future Work

� Study digital PWLCM realized in floating-pointarithmetic.

– The key is how to establish a reasonable model offloating-point arithmetic.

� Generalize the results on digital PWLCM to otherchaotic systems.

– The key is how to establish a reasonable model onhow the involved chaotic functions are calculated indigital computers.

� Other digital dynamical indicators, including the dig-itization of classical dynamical indicators.


CCCS.CityU

Outline

Related Work

Preliminaries

Major Results

Applications

Future Work

Page 53 of 53

JJ II

J I


Search

Close


Thank you!

This presentation was made with pdfwin package, andcompiled with pdflatex and postprocessed with PPower4.


CityU-PolyU Joint Seminar Series on Chaos and System ... · Future Work Page 2 of 53 JJ II J I ←-...

Documents

Transcript of CityU-PolyU Joint Seminar Series on Chaos and System ... · Future Work Page 2 of 53 JJ II J I ←-...