Topological Study and Lyapunov Exponent

of a Secure Steganographic Scheme

Jacques M. Bahi, Nicolas Friot and Christophe Guyeux

∗

Computer Science Laboratory DISC, FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comt´e, Belfort, France

Keywords:

Information Hiding, Steganography, Security, Topology, Lyapunov Exponent.

Abstract:

CI S

is a steganographic scheme proposed formerly, belonging into the small category of algorithms being

both stego and topologically secure. Due to its stego-security, this scheme is able to face attacks that take place

into the “watermark only attack” framework. Its topological security reinforce its capability to face threats in

other frameworks as “known message attack” or “known original attack”, in the Simmons’ prisoner problem.

In this research work, the study of topological properties of C I S

is enlarged by describing this scheme as

iterations over the real line, and investigating other security properties of topological nature as the Lyapunov

exponent, that have been reported as important in the ﬁeld of information hiding security. Results show that

this scheme is able to withdraw a malicious attacker in the “estimated original attack” context too.

1 INTRODUCTION

The ﬁrst fundamental work in information hiding se-

curity was realized by Cachin in the early ‘00s, in

the context of steganography (Cachin, 2004): at-

tempts of an attacker to make the distinction between

an innocent image and a stego-content was rewrit-

ten in this article as a hypothesis testing problem.

The basic properties of a stegosystem are deﬁned by

Cachin using the notions of entropy, mutual informa-

tion, and relative entropy. At the same time, Mittel-

holzer has proposed the ﬁrst theoretical framework

for analyzing the security in the second category of

algorithms studied by the information hiding commu-

nity, namely the digital watermarking (Mittelholzer,

1999). These efforts to bring a theoretical frame-

work for security in steganography and watermarking

have been followed up by Kalker, who tries to clarify

the concepts (robustness vs. security), and the clas-

siﬁcations of watermarking attacks (Kalker, 2001).

This work has been deepened by Furon et al., who

have translated Kerckhoffs’ principle (Alice and Bob

shall only rely on some previously shared secret for

privacy), from cryptography to data hiding (Furon,

2002). They used Difﬁe and Hellman methodology,

and Shannon’s cryptographic framework (Shannon,

1949), to classify the watermarking attacks into cat-

egories, according to the type of information Eve has

∗

Authors in alphabetic order

access to (Perez-Freire et al., 2006), namely: Water-

marked Only Attack (WOA), Known Message Attack

(KMA), Known Original Attack (KOA), Constant-

Message Attack (CMA), and Estimated Original At-

tacks (EOA).

Levels of security have been recently deﬁned in

these setups. The highest level of security in WOA

is called stego-security (Cayre et al., 2008), whereas

topological security tends to improve the ability to

withstand attacks in KMA, KOA, and CMA setups

(Guyeux et al., 2010). It has been previously es-

tablished that, in order to enlarge the knowledge of

the level of security of a steganographic scheme, the

quantity of disorder generated by the chaos of its

topological security can be measured evaluating the

well-known Lyapunov exponent (Mart´ınez-

Nonthe

et al., 2011; Mao and Chen, 2011; Bahi et al., 2012).

The evaluation of this exponent allow to characterize

the ability of the scheme to face an attacker in the

context of an EOA.

The ﬁrst contribution of this article consists in a

relation established between how ﬁne is a topology

and the chaotic behavior of a dynamical system de-

scribed with this topology. The second contribution is

the security study of a previously released stego and

topologically secure steganographic scheme called

CI S

, on a new topological space, namely the real

line numbers R. On this new space, the topological

security of C I S

is ﬁrstly evaluated, and its Lyapunov

exponent is then computed, in order to quantify its

275

M. Bahi J., Friot N. and Guyeux C..

Topological Study and Lyapunov Exponent of a Secure Steganographic Scheme.

DOI: 10.5220/0004504202750283

In Proceedings of the 10th International Conference on Security and Cryptography (SECRYPT-2013), pages 275-283

ISBN: 978-989-8565-73-0

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

level of disorder. Incidentally, this computation al-

lows to measure the resistance of the C I S

scheme

against a category of attacks called Estimated Origi-

nal Attack. This study follows a same canvas than a

previous work dealing with digital watermarking, but

it is conducted here for a steganographic scheme. To

achieve this work, a new semi-conjugacy model must

be written for the scheme C I S

, which is then estab-

lished and proven here.

This document is organized as follows. Notions

and ﬁrsts results concerning the mathematical theory

of chaos are introduced in the next section. Then,

in Section 3, security notions and classes of attacks

under consideration in the information hiding com-

munity are recalled. In Section 4, the steganographic

scheme studied in this document is presented, and the

formalization allowing its topological security evalu-

ation is given in the next section. This model is then

used in Section 6 to design a new semi-conjugacy al-

lowing its security study on a new space (Sect. 7),

and its Lyapunov exponent is ﬁnally evaluated in Sec-

tion 8. This paper ends by a conclusion section where

our contribution is summarized and intended future

researches are given.

2 THE MATHEMATICAL

THEORY OF CHAOS

2.1 Notations and Terminologies

In what follows, B denotes the Boolean set {0,1}, S

stands for the n

term of a sequence S, V

is for the

component of a vector V, and J0;NK is the inte-

ger interval {0,1,...,N}. Furthermore, the following

deﬁnitions will be used in this document.

Deﬁnition 1. The discrete Boolean metric is the ap-

plication δ

B −→ B deﬁned by δ(x,y) = 0 ⇔ x = y.

Deﬁnition 2. The vectorial negation is the function

−→ B

deﬁned by f



,··· ,b

N−1

)



(

,··· ,b

N−1

), where x is the negation of the Boolean

Deﬁnition 3. A strategy adapter is a sequence which

elements belong into J1,kK, where k ∈ N

∗

. The set of

all strategies having terms in J1,kK is denoted by S

Deﬁnition 4. For k ∈ N

∗

, the initial function is the

map i

−→ J1,kK deﬁned by i



)

n∈N



= S

Deﬁnition 5. Let k ∈ N

∗

. The shift function is the map

−→ S

deﬁned by σ



)

n∈N



= (S

n+1

)

n∈N

It is now possible to give some recalls in the ﬁeld

of the mathematical topology (Schwartz, 1980) to

make this document self contained.

2.2 The Chaotic Dynamical Systems

Let (X ,τ) be a topological space and f a continuous

function on (X ,τ).

Deﬁnition 6. f is said to be topologically transitive

if, for any pair of open sets U,V ⊂ X , there exists

k > 0 such that f

(U) ∩V 6= ∅.

Deﬁnition 7. (X , f) is regular if the set of periodic

points is dense in X .

It is now possible to introduce the well-established

mathematical deﬁnition of chaos (Devaney, 1989).

Deﬁnition 8. A function f

X −→ X is said to be

chaotic on X if it is regular and topologically transi-

tive.

If the topological space is indeed a metric space

(X ,d), then the sensibility of the system under iter-

ations, regarding its initial conditions, can be quanti-

ﬁed as follows.

Deﬁnition 9. f has sensitive dependence on initial

conditions if there exists δ > 0 such that, for any x ∈ X

and any neighborhood V of x, there exist y ∈ V and

n > 0 such that d



(x), f

(y)



> δ. δ is called the

constant of sensitiveness of f.

This property is implied by both the regularity and

transitivity presented above (Banks et al., 1992). And

so, when f is chaotic, fundamentally different behav-

iors are possible for the system, and they occur in an

unpredictable way.

Let us state now some basic results that surpris-

ingly cannot be found in the literature. To simplify

the presentation, some notations must be ﬁrstly intro-

duced: X

will stand for the topological space (X , τ),

whereas V

(x) is the set of neighborhoods of x for the

topology τ (in unambiguous cases, we will simply use

V (x)).

Theorem 1. Let X be a set, and τ,τ

′

two topologies on

X such that τ

′

is ﬁner than τ. Let f

X → X , continue

for both τ and τ

′

If (X

′

, f) is chaotic in the sense of Devaney, then

, f) is also chaotic.

Proof 1. Let ω

,ω

two open sets of τ. Then ω

,ω

∈

′

, because τ

′

is ﬁner than τ. As f is τ

′

−transitive,

then ∃n ∈ N,ω

∩ f

(n)

(ω

) = ∅. As a consequence,

f is τ−transitive.

Let us now establish the regularity of (X

, f), i.e.,

for all x ∈ X and all τ−neighborhood V of x, there

exists a periodic point for f in V. Let x ∈ X , and

V ∈ V

(x) a τ− neighborhood of x. By deﬁnition of

a neighborhood, ∃ω ∈ τ,x ∈ ω ⊂ V. However τ ⊂ τ

′

so ω ∈ τ

′

, and then V ∈ V

′

(x). But (X

′

, f) is regular,

So there exists a periodic point for f in V, and the

regularity of (X

, f) is proven.

SECRYPT2013-InternationalConferenceonSecurityandCryptography

276

Let us ﬁnally recall another topological quantita-

tive property of chaos:

Deﬁnition 10. A function f is said to be expansive if

∃ε > 0,∀x 6= y, ∃n ∈ N, d( f

(x), f

(y)) > ε.

Sometimes, instead of trying to prove properties

directly on the system itself, it is preferable to reduce

the initial problem to another one whose character-

istics are known or appear more accessible. Such a

reduction tool is called, in the mathematical theory of

chaos, the semi-conjugacy.

2.3 The Topological Semi-conjugacy

Deﬁnition 11. The discrete dynamical system (X , f)

is topologically semi-conjugate to the system (Y ,g)

if it exists a function ϕ

X −→ Y , both continuous

and onto, such that:

ϕ◦ f = g◦ ϕ,

that is, which makes commutative the following dia-

gram (Formenti, 1998).

−−−−→ X



Y −−−−→

In this case, the system (Y ,g) is called a factor of

the system (X , f).

Various dynamical behaviors are inherited by sys-

tems factors (Formenti, 1998). They are summarized

in the following proposition:

Proposition 1. Let (Y ,g) a factor of the system

(X , f). Then:

1. for all j 6 k, p ∈ Per

( f) =⇒ ϕ(p) ∈ Per

(g),

where Per

(h) stands for the set of points of pe-

riod n for the iteration function h.

2. (X , f) regular =⇒ (Y ,g) regular,

3. (X , f) transitive =⇒ (Y ,g) transitive.

So if (X , f) is chaotic as deﬁned by Devaney, then

(Y ,g) is chaotic too.

2.4 The Lyapunov Exponent

Some dynamical systems are very sensitive to small

changes in their initial conditions, which is illustrated

by both the constants of sensitiveness to initial condi-

tions and of expansiveness introduced respectively in

Deﬁnitions 9 and 10. However, these variations can

quickly take enormous proportions, grow exponen-

tially, and none of these constants can illustrate that.

Alexander Lyapunov has examined this phenomenon

and introduced an exponent that measures the rate at

which these small variations can grow.

Deﬁnition 12. Given f

R −→ R, the Lyapunov ex-

ponent of the system composed by x

∈ R and x

n+1

f(x

) is deﬁned by:

λ(x

) = lim

n→+∞

∑

i=1



′



i−1





Consider a dynamical system with an inﬁnitesimal

error on the initial condition x

. When the Lyapunov

exponent is positive, this error will increase exponen-

tially (situation of chaos), whereas it will decrease if

λ(x

) 6 0.

Let us now recall the information hiding security

framework developed this last decade.

3 INFORMATION HIDING

SECURITY

In the prisoner problem of Simmons (Simmons,

1984), Alice and Bob are in jail, and they want to de-

vise an escape plan by exchanging hidden messages

in innocent-looking cover contents. These messages

are to be conveyed to one another by a common war-

den, Eve, who over-drops all contents and can choose

to interrupt the communication if they appear to be

stego-contents.

In the steganography framework, in the context

of the Simmons’ prisoner problem, attacks have been

classiﬁed in (Cayre et al., 2008) as follows:

• A Watermark-Only Attack WOA occurs when an

attacker has only access to several watermarked

contents.

• A Known-Message Attack (KMA) occurs when

an attacker has access to several pairs of water-

marked contents and corresponding hidden mes-

sages.

• A Known-Original Attack KOA is when an at-

tacker has access to several pairs of watermarked

contents and their corresponding original ver-

sions.

• A Constant-Message Attack CMA occurs when

the attacker observes several watermarked con-

tents and only knows that the unknown hidden

message is the same in all contents.

• Finally, an Estimated Original Attacks (EOA) oc-

curs when the attacker has access to an estimation

of the original host signal, with possibly some es-

timation errors.

TopologicalStudyandLyapunovExponentofaSecureSteganographicScheme

277

In the framework of WOA, the stego-

security (Cayre et al., 2008) is relevant to evaluate

the security of information hiding processes. It is the

highest security level in WOA setup. To recall it, the

following notations must ﬁrstly be introduced: K is

the set of embedding keys, p(X) is the probabilistic

model of N

initial host contents, p(Y|K

) is the

probabilistic model of N

watermarked contents.

Furthermore, it is supposed in this context that each

host content has been watermarked with the same

secret key K

and the same embedding function e.

It is now possible to deﬁne the notion of stego-

security:

Deﬁnition 13 (Stego-Security). In the Watermark-

Only Attack framework, the embedding function e is

stego-secure if and only if:

∀K

∈ K, p(Y|K

) = p(X).

In the other frameworks (KOA, KMA, and CMA),

the topological security should be investigated (Friot

et al., 2011). In this article, we focus more speciﬁcally

on this topological security, which is recalled below.

To check whether an information hiding scheme

S is topologically secure or not, S must be written

as an iterate process x

n+1

= f(x

) on a metric space

(X ,d). This formulation is always possible (Bahi and

Guyeux, 2010). So,

Deﬁnition 14 (Topological Security). An information

hiding scheme S is said to be topologically secure on

(X ,d) if its iterative process has a chaotic behavior

according to Devaney.

Thus a data hiding scheme is secure if it is un-

predictable. Its iterative process must satisfy the De-

vaney’s chaos property and its level of topological se-

curity increases with the number of chaotic properties

satisﬁed by it.

This new concept of security for data hiding

schemes has been proposed in (Bahi and Guyeux,

2010) as a complementary approach to the existing

framework. It contributesto the reinforcementof con-

ﬁdence into existing secure data hiding schemes. Ad-

ditionally, the study of security in KMA, KOA, and

CMA setups is realizable in this context. Finally,

this framework can replace stego-security in situa-

tions that are not encompassed by it. In particular,

this framework is more relevant to give evaluation of

data hiding schemes claimed as chaotic.

In the EOA framework, the evaluation of the Lya-

punov exponent, which is the subject of this research

work, is relevant to quantify the level of security of

steganographic processes proven to be topologically

secure. Indeed, the Lyapunov exponent participates

to the measurement of this topological security. In an

EOA setup, the attacker has only access to estimations

of the original content. With just this knowledge, he

or she should not be in measure to recover any in-

formation about the secret message or the secret key.

The topological security, with the two other notions

of sensibility and expansiveness introduced in Deﬁ-

nitions 9 and 10, are relevant to face attacks in this

context. However, these two topological properties

give no precise quantiﬁcation of the security of the

scheme, which justiﬁes the consideration of the Lya-

punov exponent.

4 THE STEGANOGRAPHIC

SCHEME CI S

To explain how to use chaotic iterations for informa-

tion hiding, we must ﬁrstly deﬁne the signiﬁcance of

a given coefﬁcient, and the notion of most and least

signiﬁcant coefﬁcients (MSCs and LSCs).

We ﬁrst notice that terms of the original content x

that may be replaced by terms taken from the secret

message y are less important than other ones: they

could be changed without be perceived as such. For

a given host content x, MSCs are then ranks of x that

describe the relevant part of the image, whereas LSCs

translate its less signiﬁcant parts. These two deﬁni-

tions are illustrated on Figure 1, where the LSCs cor-

respond to the last three bits of each pixel.

(a) Original.

(b) MSCs. (c) LSCs (×17).

Figure 1: Most and least signiﬁcant coefﬁcients of Lena.

The steganographicscheme C I S

that generalizes

the watermarking scheme based on chaotic iterations

can now be recalled. In this part the following nota-

tions will be used: x

∈ B

is the N LSCs of a given

cover media C, m

∈ B

is the secret message to em-

SECRYPT2013-InternationalConferenceonSecurityandCryptography

278

bed into x

, S

∈ S

is the place strategy, S

∈ S

the choice strategy, and lastly S

∈ S

is the mixing

strategy.

The steganographic scheme is deﬁned by

∀(n,i, j) ∈ N

∗

× J0;N − 1K × J0;P − 1K:

(

n−1

if S

6= i

if S

= i,

and











n−1

if S

6= j

n−1

if S

= j.

where

n−1

is the Boolean negation of m

n−1

The new LSCs of the stego-content are the

Boolean vector y = x

∈ B

5 TOPOLOGICAL MODEL FOR

CI S

AND SECURITY

ANALYSIS ON X

In this section is recalled the topology used in order

to model the steganographic scheme C I S

by a dis-

crete dynamical system in a topological space (De-

vaney, 1989).

Let

J0;N − 1K × B

× J0;P − 1K × B

−→ B

(k,x,λ,m) 7−→



δ(k, j).x

δ(k, j).m



j∈J0;N−1K

where + and . are the Boolean addition and product

operations.

Consider the phase space X

deﬁned as follow:

= S

× B

× S

× B

× S

, where S

and S

are

the sets introduced in Section 2.1.

The map G

−→ X

is deﬁned by:



,x,S

,m,S





),F (i

),x, i

),m), σ

),G

(m,S

)



Then CI S

can be described by the iterations of the

following discrete dynamical system:

∈ X

and X

k+1

= G

By comparing X

and X

, it has been proven

in (Friot et al., 2011) that:

Proposition 1. X

has, at least, the cardinality of the

continuum.

A new distance has been deﬁned on X

as fol-

low: ∀X,

X ∈ X

, if X = (S

,x,S

,m,S

) and

X =

(

, ˇx,

, ˇm,

), then:

(X,

X) =

(x, ˇx) + d

(m, ˇm)+

) + d

), where:

• d

(E,

E) =

N−1

∑

k=0

δ(E

) ∈ J0;NK,

• and d

(S,

S) =

∞

∑

k=1

−

∈ [0;1]

are respectively distances on B

and S

(∀N ∈ N

∗

To demonstrate that C I S

is another example of

topological chaos in the sense of Devaney, it has been

ﬁrstly established in (Friot et al., 2011) that,

Proposition 2. G

is continuous on (X

Then it has been proven that (X

) is topologi-

cally transitive, regular, and has sensitive dependence

on initial conditions. Then we have the result:

Theorem 1. G

is a chaotic map on (X

) in the

sense of Devaney, and consequently the scheme C I S

is topologically secure.

Another theorem about the security of C I S

has

been established in (Friot et al., 2011).

Theorem 2. C IS

is stego-secure.

6 A TOPOLOGICAL

SEMI-CONJUGACY BETWEEN

AND R

In this section, by using a topologicalsemi-conjugacy,

we show that C I S

modeled by G

on X can be

described as iterations on a real interval. As our re-

searches are inspired by the work of (Bahi et al.,

2012), the proofs detailed in this document will fol-

low a same canvas. To do so, some notations and ter-

minologies must be introduced another time.

Let X

(N;P)

= S

× B

× S

× B

× S

. In what

follows and for easy understanding, we will assume

that N = 3 and P = 2. So N + P = 5 and NP

= 12.

However, an equivalent formulation of the following

can be easily obtained by replacing the bases 5 and

12 by any base (N + P) and (NP

). N has only to be

greater than P.

Deﬁnition 15. The function ψ

J1,NK × J1,PK ×

J1,PK → J0, NP

− 1K is deﬁned by: ψ





− 1)P

+ (S

− 1)P + (S

− 1).

TopologicalStudyandLyapunovExponentofaSecureSteganographicScheme

279

This function aims to convert a strategy of triplets in

a simple strategy of integers expressed in a different

base. Obviously, ψ is a bijective function, the reverse

operation will be denoted by ψ

−1

. The three projec-

tions of ψ

−1

are denoted by: ψ

−1









, ψ

−1









= S

, and

−1









= S

Table 1: Illustration of the function ψ (see Deﬁnition 15).

Base Base Base Base

N = 3 P = 2 P = 2 NP

= 12





1 1 1 0

1 1 2 1

1 2 1 2

1 2 2 3

2 1 1 4

2 1 2 5

2 2 1 6

2 2 2 7

3 1 1 8

3 1 2 9

3 2 1 10

3 2 2 11

Deﬁnition 16. Let us deﬁne ϕ

(3;2)

× B

× S

× B

× S

−→

0,2

, as fol-

lows. If



,E,S

,M,S





,. . .)

);(S

,. . .);(M

); (S

,. . .)



then ϕ



,E,S

,M,S



is the real number:

• whose integral part e is

∑

k=0

4−k

∑

k=3

4−k

k−3

, that is, the binary

digits of e are E

• whose decimal part s is equal to: s =

0,ψ













.. .

∑

+∞

k=1

−k

k−1

. s is thus expressed in base 12.

ϕ realizes the association between a point of X

(3;2)

and a real number into

0,2

. We must now trans-

late the steganographic process CI S

, which is repre-

sented by G

iterations on this real interval. To do so,

two intermediate functions over

0,2

denoted by e

and s must be introduced.

Deﬁnition 17. Let x ∈

0,2

and:

• e

,. . . , e

the binary digits of the integral part of x:

⌊x⌋ =

∑

k=0

4−k

• (s

)

k∈N

the digits of x, expressed in base 12,

where the chosen decimal decomposition of x is

the one that does not have an inﬁnite number of

11: x = ⌊x⌋ +

+∞

∑

k=0

−k−1

e and s are thus deﬁned as follows:

0,2

−→ B

× B

x 7−→ ((e

);(e

))

and

0,2

−→ J0,11K

x 7−→ (s

)

k∈N

We are now able to deﬁne the function g, whose

goal is to translate the steganographic process C I S

represented by G

on an interval of R.

Deﬁnition 18. g

0,2

−→

0,2

is such that g(x)

is the real number of

0,2

deﬁned below:

• its integral part has a binary decomposition equal

to e

′

,. . . ,e

′

, with ∀i ∈ J0,2K:

′











e(x)

if i 6= ψ

−1





e(x)

2+ψ

−1

(

)

if i = ψ

−1





and ∀i ∈ J3, 4K:

′







e(x)

if i 6= ψ

−1





e(x)

+ 1 (mod 2) if i = ψ

−1





• whose decimal part is s(x)

,s(x)

,. . .

In other words, if x =

∑

k=0

4−k

+∞

∑

k=0

−k−1

then:

g(x) =

∑

k=0

4−k





δ(k,ψ

−1

)) + 1 (mod 2)



2+ψ

−1

)



δ(k,ψ

−1

))





∑

k=3

4−k

+ δ(k,ψ

−1

) (mod 2))

+∞

∑

k=0

k+1

−k−1

SECRYPT2013-InternationalConferenceonSecurityandCryptography

280

where δ is the discrete Boolean metric introduced in

Deﬁnition 1.

Numerous metrics can be deﬁned on the set

0,2

, the most usual one being the Euclidian dis-

tance ∆(x,y) = |y− x|

. This Euclidian distance does

not reproduce exactly the notion of proximity induced

by our ﬁrst distance d

on X

. Indeed d

is richer than

∆. This is the reason why we have to introduce the

following metric.

Deﬁnition 19. Given x,y ∈

0,2

, D denotes the

function from

0,2

to R

deﬁned by: D(x,y) =



e(x),e(y)



+ D



s(x),s(y)



, where:

(e, ˇe) =

∑

k=0

δ(e

, ˇe

), and D

(s, ˇs) =

∞

∑

k=1

− ˇs

Proposition 3. D is a distance on

0,2

PROOF. The three axioms deﬁning a distance must be

checked.

• D > 0, because everything is positive in its def-

inition. If D(x,y) = 0, then D

(x,y) = 0, so the

integral parts of x and y are equal (they have

the same binary decomposition). Additionally,

(x,y) = 0, then ∀k ∈ N

∗

,s(x)

= s(y)

. In other

words, x and y have the same k−th decimal digit,

∀k ∈ N

∗

. And so x = y.

• Obviously, ∀x, y,D(x,y) = D(y,x).

• Finally, the triangle inequality is obtained due to

the fact that both δ and |x− y| satisfy it.

The convergence of sequences according to D is

not the same than the usual convergence related to the

Euclidian metric. For instance, if x

→ x according

to D, then necessarily the integral part of each x

equal to the integral part of x (at least after a given

threshold), and the decimal part of x

corresponds to

the one of x “as far as required”. D is richer and more

reﬁned than the Euclidian distance, and thus is more

precise.

ϕ has been constructed in order to be continuous

and onto, so we obtained the following theorem:

Theorem 3. The steganographic process CI S

repre-

sented by





can be considered as simple itera-

tions on R, which is illustrated by the semi-conjugacy

given below:



(3;2)



−−−−→



(3;2)







0,2



−−−−→



0,2



In other words, X

is somewhat approximately

equal to

0,2

N+P

It can be remarked that the function g is a piece-

wise linear function: it is linear on each interval hav-

ing the form



n+ 1



, n ∈ J0;2

× 12J, and its

slope is equal to 12. Let us justify these assessments:

Proposition 4. The process C I S

represented by g

deﬁned on R has derivatives of all orders on

0,2

except on the 385 points in I deﬁned by:

I =



n ∈ J0;2

× 12K



Furthermore, on each interval of the form



n+ 1



, with n ∈ J0;2

× 12J, g is a linear func-

tion having a slope equal to 12: ∀x /∈ I, g

′

(x) = 12.

PROOF. Let I



n+ 1



, with n ∈ J0;2

× 12K.

All the points of I

have the same integral part e and

the same decimal part s

: on the set I

, functions e(x)

and x 7→ s(x)

of Deﬁnition 17 only depend on n. So

all the images g(x) of these points x:

• Have the same integral part, which is e, except

probably the bit number s

. In other words, this

integer has approximately the same binary de-

composition than e, the sole exception being the

digit s

(this number is then either e + 2

12−s

e− 2

12−s

, depending on the parity of s

, i.e., it is

equal to e+ (−1)

× 2

12−s

• A shift to the left has been applied to the decimal

part y, losing by doing so the common ﬁrst digit

. In other words, y has been mapped into 12 ×

y− s

To sum up, the action of g on the points of I is as

follows: ﬁrst, make a multiplication by 12, and sec-

ond, add the same constant to each term, which is



e+ (−1)

× 2

12−s



− s

We are now able to evaluate the Lyapunov expo-

nent of our digital watermarking scheme based on

chaotic iterations, which is now described by the it-

erations on R of the g function introduced in Deﬁni-

tion 18.

7 TOPOLOGICAL SECURITY OF

CI S

ON R

According to Theorem 1, CI S

represented by the

function G

on X

is topologically secure, that is to

TopologicalStudyandLyapunovExponentofaSecureSteganographicScheme

281

say





is chaotic in the sense of Devaney. We

can deduce the same property for C I S

represented

by the g function on R for the order topology. Indeed





and



g,[0, 2

[



are semi-conjugate by ϕ

as proven in the previous section. So



g,[0, 2

[



is a chaotic system according to Devaney, because

the semi-conjugacy preserves this character (Propo-

sition 1 in Section 2.3). However the topology gen-

erated by D is ﬁner than the topology generated by

the Euclidean distance ∆, which is the order topology.

Finally, according to Theorem 1, we can afﬁrm that

the steganographic process C IS

represented by g is

chaotic in the sense of Devaney for the order topology

on R.

Having these assertions in mind, we can formulate

the following theorem:

Theorem 2 The steganographic process C I S

repre-

sented by g on R is chaotic in the sense of Devaney,

when the usual topology of R is used (the order topol-

ogy).

This result is weaker than Theorem 1, which es-

tablish the chaotic property of C I S

for a ﬁner topol-

ogy. It is as if the chaos observed using usual tools

like the Euclidian distance is still preserved when con-

sidering more powerful tools (higher resulution, i.e.,

ﬁner topologies).

The result contained in Theorem 2 is however in-

teresting, as it conﬁrms that the followed approach

does not lead to weaker properties. Indeed, this study

has taken place in a system other than the one usually

considered (X

instead of R), in order to be as closed

as possible to the ﬁnal computer machines. By do-

ing so, we prevent from any loss of chaotic properties

when computing the scheme written in mathematical

terms. However, it might be feared that the choice of

a discrete mathematics approach leads to a disorder

of lower quality. In other words, we have achieved to

prevent from a situation of great disorder lost during

the computation into machines. However, the cost of

such achievement were probably to obtain a disorder

of poor quality. Theorem 2 proves exactly the con-

trary.

8 EVALUATION OF THE

LYAPUNOV EXPONENT

Let L =



∈

0,2

∀n ∈ N,x

/∈ I



, where I is

the set of points in the real interval where g is not dif-

ferentiable (as it is explained in Proposition 4). Then,

Theorem 4. ∀x

∈ L, the Lyapunov exponent of

CI S

having x

for initial condition is equal to

λ(x

) = ln(12) > 0.

PROOF. g is piecewise linear, with a slop of 12

′

(x) = 12 where the function g is differentiable).

Then ∀x ∈ L, λ(x) = lim

n→+∞

∑

i=1



′



i−1





= lim

n→+∞

∑

i=1

ln|12| = lim

n→+∞

nln|12| =

ln12.

Remark 1. The set of initial conditions for which this

exponent is not calculable is countable. This is in-

deed the initial conditions such that an iteration value

will be a number having the form

, with n ∈ N.

Moreover, for a system having N + P cells (a number

of LSCs equal to N and a secret message to embed

of width equal to P), we will ﬁnd, mutatis mutan-

dis, an inﬁnite uncountable set of initial conditions

∈



0;2

N+P



such that λ(x

) = ln(NP

So, it is possible to make the Lyapunov exponent

of the scheme CI S

as large as possible, depending

on the number of least signiﬁcant coefﬁcients of the

cover media we decide to consider, and on the width

of the message to embed. As proven in (Guyeux et al.,

2010), a large Lyapunov exponent makes it impos-

sible to achieve the well-known “Estimated Original

Attacks” (Cayre et al., 2008).

9 CONCLUSIONS AND FUTURE

WORK

A new quantitative evaluation for the steganographic

scheme C I S

is now available: its Lyapunov expo-

nent is equal to ln(NP

), where N is the number of

least signiﬁcant coefﬁcients of the cover media and P

the width of the secret message to embed. This expo-

nent allows to quantify the ampliﬁcation of the igno-

rance on the exact initial condition (the media with-

out watermark) after several iterations of the stegano-

graphic process. It illustrates the disorder generated

by iterations of the process, reinforcing its chaotic na-

ture. Thanks to its topological security, this scheme

is already able to face an attacker in the context

of Known-Message Attack, Known-Original Attack,

and Constant-Message Attack. In addition, this result

implies that it is also able to resist in the context of an

Estimated Original Attacks.

Using the semi-conjugacy described here, it will

be possible in a future work to compare the topologi-

cal behavior of C I S

on X

and R, and to explore the

topological security of the steganography scheme us-

ing this new topology. Then, an analogue study of the

SECRYPT2013-InternationalConferenceonSecurityandCryptography

282

two other topologically secure schemes cited here will

be conducted in order to compare these processes, be-

ing thus able to choose the best one according to the

type of applications under consideration. Finally, se-

curity in steganography context will be investigated

too, and topological security will be applied in this

framework.

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