CHAOTIC ITERATIONS FOR STEGANOGRAPHY
Stego-security and Chaos-security
Nicolas Friot, Christophe Guyeux and Jacques M. Bahi
Computer Science Laboratory LIFC, University of Franche-Comt´e, 16 route de Gray, Besanc¸on, France
Keywords:
Steganography, Chaos, Security, Chaotic iterations.
Abstract:
In this paper is proposed a novel steganographic scheme based on chaotic iterations. This research work takes
place into the information hiding security fields. We show that the proposed scheme is stego-secure, which is
the highest level of security in a well defined and studied category of attack called “watermark-only attack”.
Additionally, we prove that this scheme presents topological properties so that it is one of the firsts able to
face, at least partially, an adversary when considering the others categories of attacks defined in the literature.
1 INTRODUCTION
Robustness and security are two major concerns in in-
formation hiding (Katzenbeisser and Dittmann, 2004;
Domingo-Ferrer and Bras-Amor´os, 2008). These two
concerns have been defined in (Kalker, 2001) as fol-
lows. “Robust watermarking is a mechanism to cre-
ate a communication channel that is multiplexed into
original content [...]. It is required that, firstly, the per-
ceptual degradation of the marked content [...] is min-
imal and, secondly, that the capacity of the watermark
channel degrades as a smooth function of the degra-
dation of the marked content. [...]. Watermarking se-
curity refers to the inability by unauthorized users to
have access to the raw watermarking channel. [...] to
remove, detect and estimate, write or modify the raw
watermarking bits.” We will focus in this research
work on security.
In the framework of watermarking and steganog-
raphy, security has seen several important develop-
ments since the last decade (Barni et al., 2003; Cayre
et al., 2005; Ker, 2006; Bras-Amor´os and Domingo-
Ferrer, 2008). The first fundamental work in secu-
rity was made by Cachin in the context of steganogra-
phy (Cachin, 1998). Cachin interprets the attempts of
an attacker to distinguish between an innocent image
and a stego-content as a hypothesis testing problem.
In this document, the basic properties of a stegosys-
tem are defined using the notions of entropy, mutual
information, and relative entropy. Mittelholzer, in-
spired by the work of Cachin, proposed the first theo-
retical framework for analyzing the security of a wa-
termarking scheme (Mittelholzer, 1999).
These efforts to bring a theoretical framework
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 access to (Cayre et al., 2005; Perez-Freire et al.,
2006), namely: Watermarked Only Attack (WOA),
Known Message Attack (KMA), Known Original At-
tack (KOA), and Constant-Message Attack (CMA).
Levels of security have been recently defined in these
setups. The highest level of security in WOA is called
stego-security (Cayre and Bas, 2008), whereas chaos-
security tends to improve the ability to withstand at-
tacks in KMA, KOA, and CMA setups (Guyeux et al.,
2010).
To the best of our knowledge, there exist only two
information hiding schemes that are both stego-secure
and chaos-secure (Guyeux et al., 2010). The first one
is based on a spread spectrum technique called Nat-
ural Watermarking. It is stego-secure when its pa-
rameter η is equal to 1 (Cayre and Bas, 2008). Un-
fortunately, this scheme is neither robust, nor able
to face an attacker in KOA and KMA setups, due
to its lack of a topological property called expansiv-
ity (Guyeux et al., 2010). The second scheme both
218
Friot N., Guyeux C. and M. Bahi J..
CHAOTIC ITERATIONS FOR STEGANOGRAPHY - Stego-security and Chaos-security.
DOI: 10.5220/0003545002180227
In Proceedings of the International Conference on Security and Cryptography (SECRYPT-2011), pages 218-227
ISBN: 978-989-8425-71-3
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

chaos-secure and stego-secure is based on chaotic it-
erations (Bahi and Guyeux, 2010b). However, it al-
lows to embed securely only one bit per embedding
parameters. The objective of this research work is to
improvethe scheme presented by authors of (Bahi and
Guyeux, 2010b), in such a way that more than one bit
can be embedded.
The remainder of this document is organized as
follows. In Section 2, some basic recalls concerning
both chaotic iterations and Devaney’s chaos are given.
In Section 3 are presented results and information hid-
ing scheme on which our work is based. Classes of at-
tacks considered in this paper are detailed in Section
4. Stego-security and chaos-security are recalled too
in this section. The new information hiding scheme
is given in Section 5. Its stego-security is studied
in the next section. The topological framework mak-
ing it possible to evaluate chaos-security is introduced
in Section 7. Then the topological properties of our
scheme are investigated in the next section, leading
to the evaluation of its chaos-security. This research
work ends by a conclusion section where our contri-
bution is summarized and intended future researches
are presented.
2 BASIC RECALLS
2.1 Chaotic Iterations
In the sequel S
n
denotes the n
th
term of a sequence S
and V
i
is for the i
th
component of a vector V. Finally,
the following notation is used: J0;NK = {0, 1,...,N}.
Let us consider a system of a finite number N of
elements (or cells), so that each cell has a boolean
state. A sequence of length N of boolean states of the
cells corresponds to a particular state of the system. A
sequence that elements belong into J0;N − 1K is called
a strategy. The set of all strategies is denoted by S.
Definition 1. The set B denoting {0,1}, let f :
B
N
−→ B
N
be a function and S ∈ S be a strategy. The
so-called chaotic iterations are defined by x
0
∈ B
N
and ∀(n,i) ∈ N
∗
× J0;N − 1K:
x
n
i
=

x
n−1
i
if S
n
6= i,

f(x
n−1
)

S
n
if S
n
= i.
2.2 Devaney’s Chaotic Dynamical
Systems
Some topological definitions and properties taken
from the mathematical theory of chaos are recalled
in this section.
Let (X , d) be a metric space and f a continuous
function on (X ,d).
Definition 2. f is said to be topologically transitiveif,
for any pair of open sets U,V ⊂ X , there exists k > 0
such that f
k
(U) ∩V 6= ∅.
Definition 3. (X , f ) is said to be regular if the set of
periodic points is dense in X .
Definition 4. f has sensitive dependence on initial
conditionsif 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 | f
n
(x) − f
n
(y)| > δ.
δ is called the constant of sensitivity of f.
It is now possible to introduce the well-established
mathematical definition of chaos (Devaney, 1989),
Definition 5. A function f : X −→ X is said to be
chaotic on X if:
1. f is regular,
2. f is topologically transitive,
3. f has sensitive dependence on initial conditions.
When f is chaotic, then the system (X , f) is
chaotic and quoting Devaney: “it is unpredictable be-
cause of the sensitive dependence on initial condi-
tions. It cannot be broken down or simplified into two
subsystems which do not interact because of topo-
logical transitivity. And in the midst of this random
behavior, we nevertheless have an element of regu-
larity”. Fundamentally different behaviors are conse-
quently possible and occur in an unpredictable way.
Let us finally remark that,
Theorem 1 ((Banks et al., 1992)). If a function is
regular and topologicaly transitive on a metric space,
then the function is sensitive on initial conditions.
3 INFORMATION HIDING BASED
ON CHAOTIC ITERATIONS
3.1 Topology of Chaotic Iterations
In this section, we give the outline proofs establish-
ing the topological properties of chaotic iterations. As
our scheme is inspired by the work of Guyeux et al.
(Guyeux et al., 2010; Bahi and Guyeux, 2010b; Bahi
and Guyeux, 2010a), the proofs detailed at the end of
this document will follow a same canvas.
Let us firstly introduce some notations and termi-
nologies.
Definition 6. Let k ∈ N
∗
. A strategy adapter is a se-
quence which elements belong into J0, k − 1K. The set
of all strategies with terms in J0,k − 1K is denoted by
S
k
.
CHAOTIC ITERATIONS FOR STEGANOGRAPHY - Stego-security and Chaos-security
219

Definition 7. The discrete boolean metric is the ap-
plication δ : B −→ B defined by δ(x,y) = 0 ⇔ x = y.
Definition 8. Let k ∈ N
∗
. The initial function is the
map i
k
defined by:
i
k
: S
k
−→ J0,k − 1K
(S
n
)
n∈N
7−→ S
0
Definition 9. Let k ∈ N
∗
. The shift function is the
map σ
k
defined by:
σ
k
: S
k
−→ S
k
(S
n
)
n∈N
7−→ (S
n+1
)
n∈N
Definition 10. Given a function f : B
N
→ B
N
, the
function F
f
is defined by:
F
f
: J0;N −1K × B
N
−→ B
N
(k, E) 7−→

E
j
.δ(k, j) + f (E)
k
.δ(k, j)

j∈J0;N−1K
Definition 11. The phase space used for chaotic iter-
ations is denoted by X
1
and defined by X
1
= S
N
×B
N
.
Definition 12. Given a function f : B
N
→ B
N
, the
map G
f
is defined by:
G
f
: X
1
−→ X
1
(S,E) 7−→ (σ
N
(S),F
f
(i
N
(S),E))
With these definitions, chaotic iterations can be
described by the following iterations of the discret dy-
namical system:

X
0
∈ X
1
∀k ∈ N
∗
,X
k+1
= G
f
(X
k
)
Finally, a new distance d
1
between two points has
been defined by:
Definition 13 (Distance d
1
on X
1
). ∀(S,E),(
ˇ
S,
ˇ
E) ∈
X
1
, d
1
((S,E);(
ˇ
S,
ˇ
E)) = d
B
N
(E,
ˇ
E) + d
S
N
(S,
ˇ
S),
where:
• d
B
N
(E,
ˇ
E) =
N−1
∑
k=0
δ(E
k
,
ˇ
E
k
) ∈ J0;NK
• d
S
N
(S,
ˇ
S) =
9
N
∞
∑
k=1
|S
k
−
ˇ
S
k
|
10
k
∈ [0;1].
are respectively two distances on B
N
and S
N
(∀N ∈
N
∗
).
Remark 1. This new distance has been introduced
by authors of (Bahi and Guyeux, 2010a) to satisfy
the following requirements. When the number of dif-
ferent cells between two systems is increasing, then
their distance should increase too. In addition, if two
systems present the same cells and their respective
strategies start with the same terms, then the distance
between these two points must be small, because the
evolution of the two systems will be the same for a
while. The distance presented above follows these
recommendations.
It is then proven that,
Proposition 1. G
f
is a continuous function on
(X
1
,d
1
), for all f : B
N
→ B
N
.
Let us now recall the iteration function used by
authors of (Bahi and Guyeux, 2010b).
Definition 14. The vectorial negation is the function
defined by:
f
0
: B
N
−→ B
N
(b
0
,·· · , b
N−1
) 7−→ (b
0
,·· · , b
N−1
)
In the metric space (X
1
,d
1
), G
f
0
satisfies the three
conditions for Devaney’s chaos: regularity, transitiv-
ity, and sensitivity. So,
Theorem 2. G
f
0
is a chaotic map on (X
1
,d
1
) accord-
ing to Devaney.
Finally, it has been stated in (Bahi and Guyeux,
2010a) that,
Proposition 2. The phase space X
1
has, at least, the
cardinality of the continuum.
3.2 Chaotic Iterations for Data Hiding
To explain how to use chaotic iterations for informa-
tion hiding, we must firstly define the significance of
a given coefficient.
3.2.1 Most and Least Significant Coefficients
We first notice that terms of the original content x that
may be replaced by terms issued from the watermark
y are less important than other: they could be changed
without be perceived as such. More generally, a signi-
fication function attaches a weight to each term defin-
ing a digital media, depending on its position t.
Definition 15. A signification function is a real se-
quence (u
k
)
k∈N
.
Example 1. Let us consider a set of grayscale images
stored into portable graymap format (P3-PGM): each
pixel ranges between 256 gray levels, i.e., is mem-
orized with eight bits. In that context, we consider
u
k
= 8 − (k mod 8) to be the k-th term of a signi-
fication function (u
k
)
k∈N
. Intuitively, in each group
of eight bits (i.e., for each pixel) the first bit has an
importance equal to 8, whereas the last bit has an im-
portance equal to 1. This is compliant with the idea
that changing the first bit affects more the image than
changing the last one.
Definition 16. Let (u
k
)
k∈N
be a signification func-
tion, m and M be two reals s.t. m < M.
• The most significant coefficients (MSCs) of x is
the finite vector
u
M
=

k


k ∈ N and u
k
> M and k ≤| x |

;
SECRYPT 2011 - International Conference on Security and Cryptography
220

• The least significant coefficients (LSCs) of x is the
finite vector
u
m
=

k


k ∈ N and u
k
≤ m and k ≤| x |

;
• The passive coefficients of x is the finite vector
u
p
=

k


k ∈ N and u
k
∈]m;M[ and k ≤| x |

.
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
definitions are illustrated on Figure 1, where the sig-
nificance function (u
k
) is defined as in Example 1,
M = 5, and m = 6.
(a) Original Lena.
(b) MSCs of Lena. (c) LSCs of Lena (×17).
Figure 1: Most and least significant coefficients of Lena.
3.2.2 Presentation of the Scheme
Authors of (Bahi and Guyeux, 2010b) have proposed
to use chaotic iterations as an information hiding
scheme, as follows. Let:
• (K,N) ∈ [0;1] × N be an embedding key,
• X ∈ B
N
be the N LSCs of a cover C,
• (S
n
)
n∈N
∈ J0,N − 1K
N
be a strategy, which de-
pends on the message to hide M ∈ [0;1] and K,
• f
0
: B
N
→ B
N
be the vectorial logical negation.
So the watermarked media is C whose LSCs are
replaced by Y
K
= X
N
, where:

X
0
= X
∀n < N,X
n+1
= G
f
0
(X
n
).
Two ways to generate (S
n
)
n∈N
are given by these
authors, namely Chaotic Iterations with Independent
Strategy (CIIS) and Chaotic Iterations with Depen-
dent Strategy (CIDS). In CIIS, the strategy is inde-
pendent from the cover media C, whereas in CIDS the
strategy will be dependent on C. As we will use the
CIIS strategy in this document, we recall it below. Fi-
nally, MSCs are not used here, as we do not consider
the case of authenticated watermarking.
3.2.3 CIIS Strategy
Let us firstly give the definition of the Piecewise Lin-
ear Chaotic Map (PLCM, see (Shujun et al., 2001)):
F(x, p) =



x/p if x ∈ [0; p],
(x− p)/(
1
2
− p) if x ∈

p;
1
2

,
F(1 − x, p) else,
where p ∈

0;
1
2

is a “control parameter”.
Then, the general term of the strategy (S
n
)
n
in
CIIS setup is defined by the following expression:
S
n
= ⌊N × K
n
⌋ + 1, where:



p ∈

0;
1
2

K
0
= M ⊗ K
K
n+1
= F(K
n
, p),∀n ≤ N
0
in which ⊗ denotes the bitwise exclusive or (XOR)
between two floating part numbers (i.e., between their
binary digits representation).
4 DATA HIDING SECURITY
4.1 Classification of Attacks
In the steganography framework, attacks have been
classified in (Cayre and Bas, 2008) as follows.
Definition 17. Watermark-Only Attack (WOA) occurs
when an attacker has only access to several water-
marked contents.
Definition 18. Known-Message Attack (KMA) occurs
when an attacker has access to several pairs of water-
marked contents and corresponding hidden messages.
Definition 19. Known-Original Attack (KOA) is when
an attacker has access to several pairs of water-
marked contents and their corresponding original
versions.
Definition 20. Constant-Message Attack (CMA) oc-
curs when the attacker observes several watermarked
contents and only knows that the unknown hidden
message is the same in all contents.
CHAOTIC ITERATIONS FOR STEGANOGRAPHY - Stego-security and Chaos-security
221

4.2 Stego-Security
In the prisoner problem of Simmons (Simmons, 1984;
Bergmair and Katzenbeisser, 2006), Alice and Bob
are in jail, and they want to, possibly, devise an escape
plan by exchanging hidden messages in innocent-
looking covercontents. These messages are to be con-
veyed to one another by a common warden, Eve, who
over-drops all contents and can choose to interrupt the
communication if they appear to be stego-contents.
The stego-security, defined in this framework, is
the highest security level in WOA setup (Cayre and
Bas, 2008). To recall it, we need the following nota-
tions:
• K is the set of embedding keys,
• p(X) is the probabilistic model of N
0
initial host
contents,
• p(Y|K
1
) is the probabilistic model of N
0
water-
marked contents.
Furthermore, it is supposed in this context that
each host content has been watermarked with the
same secret key K
1
and the same embedding function
e.
It is now possible to define the notion of stego-
security:
Definition 21 (Stego-Security). The embedding func-
tion e is stego-secure if and only if:
∀K
1
∈ K, p(Y|K
1
) = p(X).
To the best of our knowledge, until now, only two
schemes have been proven to be stego-secure. On the
one hand, the authors of (Cayre and Bas, 2008) have
established that the spread spectrum technique called
Natural Watermarking is stego-secure when its distor-
tion parameter η is equal to 1. On the other hand, it
has been proven in (Guyeux et al., 2010) that:
Proposition 3. Chaotic Iterations with Independent
Strategy (CIIS) are stego-secure.
4.3 Chaos-Security
To check whether an information hiding scheme S is
chaos-secure or not, S must be written as an iterate
process x
n+1
= f(x
n
) on a metric space (X ,d). This
formulation is always possible (Bahi and Guyeux,
2010c). So,
Definition 22 (Chaos-Security). An information hid-
ing scheme S is said to be chaos-secure on (X ,d) if
its iterative process has a chaotic behavior according
to Devaney.
In the approach presented by Guyeux et al., a data
hiding scheme is secure if it is unpredictable. Its iter-
ative process must satisfy the Devaney’s chaos prop-
erty and its level of chaos-security 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,
2010c) as a complementary approach to the existing
framework. It contributesto the reinforcement of 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.
5 THE IMPROVED ALGORITHM
In this section is introduced a new algorithm that gen-
eralize the scheme presented by authors of (Bahi and
Guyeux, 2010b).
Let us firstly introduce the following notations:
• x
0
∈ B
N
is the N least significant coefficients of a
given cover media C.
• m
0
∈ B
P
is the watermark to embed into x
0
.
• S
1
∈ S
N
is a strategy called place strategy.
• S
2
∈ S
P
is a strategy called choice strategy.
• Lastly, S
3
∈ S
P
is a strategy called mixing strat-
egy.
Our information hiding scheme called Steganog-
raphy by Chaotic Iterations and Substitution with
Mixing Message (SCISMM) is defined by ∀(n,i, j) ∈
N
∗
× J0;N − 1K × J0;P − 1K:

















x
n
i
=

x
n−1
i
if S
n
1
6= i
m
S
n
2
if S
n
1
= i.
m
n
j
=





m
n−1
j
if S
n
3
6= j
m
n−1
j
if S
n
3
= j.
where m
n−1
j
is the boolean negation of m
n−1
j
.
The stego-content is the boolean vector y = x
P
∈
B
N
.
6 STUDY OF STEGO-SECURITY
Let us prove that,
SECRYPT 2011 - International Conference on Security and Cryptography
222

Proposition 4. SCISMM is stego-secure.
Proof. Let us suppose that x
0
∼ U

B
N

and m
0
∼
U

B
P

in a SCISMM setup. We will proveby a math-
ematical induction that ∀n ∈ N,x
n
∼ U

B
N

. The
base case is obvious according to the uniform reparti-
tion hypothesis.
Let us now suppose that the statement x
n
∼
U

B
N

holds for some n. For a given k ∈ B
N
, we de-
note by
˜
k
i
∈ B
N
the vector defined by: ∀i ∈ J0;N − 1K,
if k = (k
0
,k
1
,.. .,k
i
,.. .,k
N−2
,k
N−1
),
then
˜
k
i
=

k
0
,k
1
,.. .,k
i
,.. .,k
N−2
,k
N−1

.
Let E
i, j
be the following events:
∀(i, j) ∈ J0;N − 1K × J0;P − 1K,E
i, j
=
S
n+1
1
= i∧ S
n+1
2
= j ∧ m
n+1
j
= k
i
∧

x
n
= k∨ x
n
=
˜
k
i

,
and p = P

x
n+1
= k

. So,
p = P


_
i∈J0;N−1K, j∈J0;P−1K
E
i, j


.
We now introduce the following notation: P
1
(i) =
P

S
n+1
1
= i

, P
2
( j) = P

S
n+1
2
= j

, P
3
(i, j) =
P

m
n+1
j
= k
i

, and P
4
(i) = P

x
n
= k∨ x
n
=
˜
k
i

.
These four events are independent in SCISMM
setup, thus:
p =
∑
i∈J0;N−1K, j∈J0;P−1K
P
1
(i)P
2
(i)P
3
(i, j)P
4
(i).
According to Proposition 3, P

m
n+1
j
= k
i

=
1
2
. As
the two events are incompatible:
P

x
n
= k∨ x
n
=
˜
k
i

= P(x
n
= k) + P

x
n
=
˜
k
i

.
Then, by using the inductive hypothesis:
P(x
n
= k) =
1
2
N
, and P

x
n
=
˜
k
i

=
1
2
N
.
Let S be defined by
S =
∑
i∈J0;N−1K, j∈J0;P−1K
P
1
(i)P
2
( j).
Then p = 2×
1
2
×
1
2
N
× S =
1
2
N
× S.
S can now be evaluated:
S =
∑
i∈J0;N−1K, j∈J0;P−1K
P
1
(i)P
2
( j)
=
∑
i∈J0;N−1K
P
1
(i) ×
∑
j∈J0;P−1K
P
2
( j).
The set of events

S
n+1
1
= i

for i ∈ J0;N − 1K and
the set of events

S
n+1
2
= j

for j ∈ J0;P−1K are both
a partition of the universe of possible, so S = 1.
Finally, P

x
n+1
= k

=
1
2
N
, which leads to x
n+1
∼
U

B
N

. This result is true ∀n ∈ N, we thus have
proven that the stego-content y is uniform in the set
of possible stego-content, so y ∼ U

B
N

when x ∼
U

B
N

.
7 TOPOLOGICAL MODEL
In this section, we prove that SCISMM can be mod-
eled as a discret dynamical system in a topological
space. We will show in the next section that SCISMM
is a case of topological chaos in the sense of Devaney.
7.1 Iteration Function and Phase Space
Let
F : J0;N − 1K × B
N
× J0;P − 1K × B
P
−→ B
N
(k, x,λ,m) 7−→

δ(k, j).x
j
+ δ(k, j).m
λ

j∈J0;N−1K
where + and . are the boolean addition and product
operations.
Consider the phase space X
2
defined as follow:
X
2
= S
N
× B
N
× S
P
× B
P
× S
P
,
where S
N
and S
P
are the sets introduced in Section 5.
We define the map G
f
0
: X
2
−→ X
2
by:
G
f
0
(S
1
,x,S
2
,m,S
3
) =
(σ
N
(S
1
),F(i
N
(S
1
),x,i
P
(S
2
),m),σ
P
(S
2
),G
f
0
(m,S
3
),σ
P
(S
3
))
Then SCISMM can be described by the iterations of
the following discret dynamical system:

X
0
∈ X
2
X
k+1
= G
f
0
(X
k
).
7.2 Cardinality of X
2
By comparing X
2
and X
1
, we have the following re-
sult.
Proposition 5. The phase space X
2
has, at least, the
cardinality of the continuum.
Proof. Let ϕ be the map defined as follow:
ϕ : X
1
−→ X
2
(S,x) 7−→ (S,x,0,0, 0)
ϕ is injective. So the cardinality of X
2
is greater than
or equal to the cardinality of X
1
. And consequently
X
2
has at least the cardinality of the continuum.
Remark 2. This result is independent on the number
of cells of the system.
7.3 A New Distance on X
2
We define a newdistance on X
2
as follow: ∀X,
ˇ
X ∈ X
2
,
if X = (S
1
,x,S
2
,m,S
3
) and
ˇ
X = (
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
), then:
d
2
(X,
ˇ
X) = d
B
N
(x, ˇx) + d
B
P
(m, ˇm)
+ d
S
N
(S
1
,
ˇ
S
1
) +d
S
P
(S
2
,
ˇ
S
2
) +d
S
P
(S
3
,
ˇ
S
3
),
where d
B
N
, d
B
P
, d
S
N
, and d
S
P
are the same distances
than in Definition 13.
CHAOTIC ITERATIONS FOR STEGANOGRAPHY - Stego-security and Chaos-security
223

7.4 Continuity of SCISMM
To prove that SCISMM is another example of topo-
logical chaos in the sense of Devaney, G
f
0
must be
continuous on the metric space (X
2
,d
2
).
Proposition 6. G
f
0
is a continuous function on
(X
2
,d
2
).
Proof. We use the sequential continuity.
Let ((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
)
n∈N
be a se-
quence of the phase space X
2
, which con-
verges to (S
1
,x,S
2
,m,S
3
). We will prove that
(G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
))
n∈N
converges to
G
f
0
(S
1
,x,S
2
,m,S
3
). Let us recall that for all n, (S
1
)
n
,
(S
2
)
n
and (S
3
)
n
are strategies, thus we consider a
sequence of strategies (i.e., a sequence of sequences).
As d
2
(((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
),(S
1
,x,S
2
,m,S
3
))
converges to 0, each distance d
B
N
(x
n
,x), d
B
P
(m
n
,m),
d
S
N
((S
1
)
n
,S
1
), d
S
P
((S
2
)
n
,S
2
), and d
S
P
((S
3
)
n
,S
3
)
converges to 0. But d
B
N
(x
n
,x) and d
B
P
(m
n
,m) are
integers, so ∃n
0
∈ N,∀n > n
0
,d
B
N
(x
n
,x) = 0 and
∃n
1
∈ N,∀n > n
1
,d
B
P
(m
n
,m) = 0.
Let n
3
= Max(n
0
,n
1
). In other words, there exists
a threshold n
3
∈ N after which no cell will change its
state: ∃n
3
∈ N,n > n
3
=⇒ (x
n
= x) ∧ (m
n
= m).
In addition, d
S
N
((S
1
)
n
,S
1
) −→ 0,
d
S
P
((S
2
)
n
,S
2
) −→ 0, and d
S
P
((S
3
)
n
,S
3
) −→ 0,
so ∃n
4
,n
5
,n
6
∈ N,
• ∀n > n
4
,d
S
N
((S
1
)
n
,S
1
) < 10
−1
,
• ∀n > n
5
,d
S
P
((S
2
)
n
,S
2
) < 10
−1
,
• ∀n > n
6
,d
S
P
((S
3
)
n
,S
3
) < 10
−1
.
Let n
7
= Max(n
4
,n
5
,n
6
). For n > n
7
, all the strate-
gies (S
1
)
n
, (S
2
)
n
, and (S
3
)
n
have the same first term,
which are respectively(S
1
)
0
,(S
2
)
0
and (S
3
)
0
:∀n > n
7
,
((S
1
)
n
0
= (S
1
)
0
) ∧ ((S
2
)
n
0
= (S
2
)
0
) ∧ ((S
3
)
n
0
= (S
3
)
0
).
Let n
8
= Max(n
3
,n
7
). After the n
8
−th term, states
of x
n
and x on the one hand, and m
n
and m on the other
hand, are identical. Additionally, strategies (S
1
)
n
and
S
1
, (S
2
)
n
and S
2
, and (S
3
)
n
and S
3
start with the same
first term.
Consequently, states of
G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
) and
G
f
0
(S
1
,x,S
2
,m,S
3
) are equal, so, after the (n
8
)
th
term, the distance d
2
between these two points is
strictly smaller than 3.10
−1
, so strictly smaller than
1.
We now prove that the distance be-
tween (G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
)) and
(G
f
0
(S
1
,x,S
2
,m,S
3
)) is convergent to 0. Let
ε > 0.
• If ε > 1, we have seen that distance be-
tween G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
) and
G
f
0
(S
1
,x,S
2
,m,S
3
) is strictly less than 1 af-
ter the (n
8
)
th
term (same state).
• If ε < 1, then ∃k ∈ N,10
−k
>
ε
3
> 10
−(k+1)
. As
d
S
N
((S
1
)
n
,S
1
), d
S
P
((S
2
)
n
,S
2
) and d
S
P
((S
3
)
n
,S
3
)
converges to 0, we have:
– ∃n
9
∈ N,∀n > n
9
,d
S
N
((S
1
)
n
,S
1
) < 10
−(k+2)
,
– ∃n
10
∈ N,∀n > n
10
,d
S
P
((S
2
)
n
,S
2
) < 10
−(k+2)
,
– ∃n
11
∈ N,∀n > n
11
,d
S
P
((S
3
)
n
,S
3
) < 10
−(k+2)
.
Let n
12
= Max(n
9
,n
10
,n
11
) thus after n
12
, the k+
2 first terms of (S
1
)
n
and S
1
, (S
2
)
n
and S
2
, and
(S
3
)
n
and S
3
, are equal.
As a consequence, the k + 1 first entries of
the strategies of G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
) and
G
f
0
(S
1
,x,S
2
,m,S
3
) are the same (due to the shift of
strategies) and following the definition of d
S
N
and
d
S
P
:
d
2
(G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
);G
f
0
(S
1
,x,S
2
,m,S
3
))
is equal to :
d
S
N
((S
1
)
n
,S
1
) + d
S
P
((S
2
)
n
,S
2
) + d
S
P
((S
3
)
n
,S
3
)
which is smaller than 3.10
−(k+1)
6 3.
ε
3
= ε.
Let N
0
= max(n
8
,n
12
). We can claim that
∀ε > 0,∃N
0
∈ N,∀n > N
0
,
d
2
(G
f
0
((S
1
)
n
,x
n
,(S
2
)
n
,m
n
,(S
3
)
n
);G
f
0
(S
1
,x,S
2
,m,S
3
)) 6 ε.
G
f
0
is consequently continuous on (X
2
,d
2
).
8 SCISMM IS CHAOTIC
To prove that we are in the framework of Devaney’s
topological chaos, we have to check the regularity,
transitivity, and sensitivity conditions.
8.1 Regularity
Proposition 7. Periodic points of G
f
0
are dense in X
2
.
Proof. Let (
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
) ∈ X
2
and ε > 0. We are
looking for a periodic point (
e
S
1
, ex,
e
S
2
, em,
e
S
3
) satisfying
d
2
((
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
);(
e
S
1
, ex,
e
S
2
, em,
e
S
3
)) < ε.
As ε can be strictly lesser than 1, we must
choose ex = ˇx and em = ˇm. Let us define k
0
(ε) =
⌊log
10
(
ε
3
)⌋ + 1 and consider the set: S
ˇ
S
1
,
ˇ
S
2
,
ˇ
S
3
,k
0
(ε)
=
n
S ∈ S
N
× S
P
× S
P
/((S
1
)
k
=
ˇ
S
1
k
) ∧ ((S
2
)
k
=
ˇ
S
2
k
))
∧((S
3
)
k
=
ˇ
S
3
k
)),∀k 6 k
0
(ε)
o
.
SECRYPT 2011 - International Conference on Security and Cryptography
224

Then, ∀(S
1
,S
2
,S
3
) ∈ S
ˇ
S
1
,
ˇ
S
2
,
ˇ
S
3
,k
0
(ε)
,
d
2
((S
1
, ˇx,S
2
, ˇm,S
3
);(
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
)) < 3.
ε
3
= ε.
It remains to choose (
e
S
1
,
e
S
1
,
e
S
1
) ∈ S
ˇ
S
1
,
ˇ
S
2
,
ˇ
S
3
,k
0
(ε)
such
that (
e
S
1
, ex,
e
S
2
, em,
e
S
3
) = (
e
S
1
, ˇx,
e
S
2
, ˇm,
e
S
3
) is a periodic
point for G
f
0
.
Let J = {i ∈ J0;N − 1K/x
i
6= ˇx
i
, where
(S
1
,x,S
2
,m,S
3
) = G
k
0
f
0
(
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
)
o
,
λ = card(J ), and j
0
< j
1
< ... < j
λ−1
the ele-
ments of J .
1. Let us firstly build three strategies: S
∗
1
, S
∗
2
, and S
∗
3
,
as follows.
(a) (S
∗
1
)
k
=
ˇ
S
1
k
, (S
∗
2
)
k
=
ˇ
S
2
k
, and (S
∗
3
)
k
=
ˇ
S
3
k
, if
k 6 k
0
(ε).
(b) Let us now explain how to replace ˇx
j
q
, ∀q ∈
J0;λ − 1K:
First of all, we must replace ˇx
j
0
:
i. If ∃λ
0
∈ J0;P − 1K/ ˇx
j
0
= m
λ
0
, then we
can choose (S
∗
1
)
k
0
+1
= j
0
, (S
∗
2
)
k
0
+1
= λ
0
,
(S
∗
3
)
k
0
+1
= λ
0
, and so I
j
0
will be equal to 1.
ii. If such a λ
0
does not exist, we choose:
(S
∗
1
)
k
0
+1
= j
0
, (S
∗
2
)
k
0
+1
= 0, (S
∗
3
)
k
0
+1
= 0,
(S
∗
1
)
k
0
+2
= j
0
, (S
∗
2
)
k
0
+2
= 0, (S
∗
3
)
k
0
+2
= 0,
and I
j
0
= 2.
All of the ˇx
j
q
are replaced similarly. The other
terms of S
∗
1
, S
∗
2
, and S
∗
3
are constructed iden-
tically, and the values of I
j
q
are defined in the
same way.
Let γ =
∑
λ−1
q=0
I
j
q
.
(c) Finally, let (S
∗
1
)
k
= (S
∗
1
)
j
, (S
∗
2
)
k
= (S
∗
2
)
j
, and
(S
∗
3
)
k
= (S
∗
3
)
j
, where j 6 k
0
(ε)+ γ is satisfying
j ≡ k [mod (k
0
(ε) + γ)], if k > k
0
(ε) + γ.
So, G
k
0
(ε)+γ
f
0
(S
∗
1
, ˇx,S
∗
2
, ˇm, S
∗
3
) = (S
∗
1
, ˇx,S
∗
2
,m,S
∗
3
).
Let K = {i ∈ J0;P − 1K/m
i
6= ˇm
i
, where
G
k
0
(ε)+γ
f
0
(S
∗
1
, ˇx,S
∗
2
, ˇm,S
∗
3
) = (S
∗
1
, ˇx,S
∗
2
,m,S
∗
3
)
o
,
µ = card(K ), and r
0
< r
1
< ... < r
µ−1
the ele-
ments of K .
2. Let us now build the strategies
e
S
1
,
e
S
2
,
e
S
3
.
(a) Firstly, let
e
S
1
k
= (S
∗
1
)
k
,
e
S
2
k
= (S
∗
2
)
k
, and
e
S
3
k
=
(S
∗
3
)
k
, if k 6 k
0
(ε) + γ.
(b) How to replace ˇm
r
q
,∀q ∈ J0;µ − 1K:
First of all, let us explain how to replace ˇm
r
0
:
i. If ∃µ
0
∈ J0;N − 1K/ˇx
µ
0
= m
r
0
, then we
can choose
e
S
1
k
0
+γ+1
= µ
0
,
e
S
2
k
0
+γ+1
= r
0
,
e
S
2
k
0
+γ+1
= r
0
.
In that situation, we define J
r
0
= 1.
ii. If such a µ
0
does not exist, then we can choose:
e
S
1
k
0
+γ+1
= 0,
e
S
2
k
0
+γ+1
= r
0
,
e
S
2
k
0
+γ+1
= r
0
,
e
S
1
k
0
+γ+2
= 0,
e
S
2
k
0
+γ+2
= r
0
,
e
S
2
k
0
+γ+2
= 0,
e
S
1
k
0
+γ+3
= 0,
e
S
2
k
0
+γ+3
= r
0
,
e
S
2
k
0
+γ+3
= 0.
Let J
r
0
= 3.
Then the other ˇm
r
q
are replaced as previously,
the other terms of
e
S
1
,
e
S
2
, and
e
S
3
are con-
structed in the same way, and the values of J
r
q
are defined similarly.
Let α =
∑
µ−1
q=0
J
r
q
.
(c) Finally, let
e
S
1
k
=
e
S
1
j
,
e
S
2
k
=
e
S
2
j
, and
e
S
3
k
=
e
S
3
j
where j 6 k
0
(ε) + γ + α is satisfying j ≡
k [mod (k
0
(ε) + γ+ α)], if k > k
0
(ε) + γ+ α.
So, G
k
0
(ε)+γ+α
f
0
(
e
S
1
, ˇx,
e
S
2
, ˇm,
e
S
3
) = (
e
S
1
, ˇx,
e
S
2
, ˇm,
e
S
3
)
Then, (
e
S
1
,
e
S
2
,
e
S
3
) ∈ S
ˇ
S
1
,
ˇ
S
2
,
ˇ
S
3
,k
0
(ε)
defined as pre-
vious is such that (
e
S
3
, ˇx,
e
S
3
, ˇm,
e
S
3
) is a periodic
point, of period k
0
(ε) + γ + α, which is ε−close to
(
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
).
As a conclusion, (X
2
,G
f
0
) is regular.
8.2 Transitivity
Proposition 8. (X
2
,G
f
0
) is topologically transitive.
Proof. Let us define X : X
2
→ B
N
, such that
X (S
1
,x,S
2
,m,S
3
) = x and M : X
2
→ B
P
,
such that M (S
1
,x,S
2
,m,S
3
) = m. Let
B
A
= B (X
A
,r
A
) and B
B
= B (X
B
,r
B
)
be two open balls of X
2
, with
X
A
= ((S
1
)
A
,x
A
,(S
2
)
A
,m
A
,(S
3
)
A
) and X
B
=
((S
1
)
B
,x
B
,(S
2
)
B
,m
B
,(S
3
)
B
). We are look-
ing for
e
X = (
e
S
1
, ex,
e
S
2
, em,
e
S
3
) in B
A
such that
∃n
0
∈ N,G
n
0
f
0
(
e
X) ∈ B
B
.
e
X must be in B
A
and r
A
can be strictly lesser than 1, so
ex = x
A
and em = m
A
. Let k
0
= ⌊log
10
(
r
A
3
) + 1⌋. Let us
notice S
X
A
,k
0
=

(S
1
,S
2
,S
3
) ∈ S
N
× (S
P
)
2
/∀k 6 k
0
,
(S
k
1
= (S
1
)
k
A
) ∧ (S
k
2
= (S
2
)
k
A
) ∧ (S
k
3
= (S
3
)
k
A
))

.
Then ∀(S
1
,S
2
,S
3
) ∈ S
X
A
,k
0
,(S
1
, ex,S
2
, em,S
3
) ∈ B
A
.
Let J = {i ∈ J0,N − 1K/ ˇx
i
6= X (X
B
)
i
, where
(
ˇ
S
1
, ˇx,
ˇ
S
2
, ˇm,
ˇ
S
3
) = G
k
0
f
0
(X
A
)
o
, λ = card(J ),
and j
0
< j
1
< ... < j
λ−1
the elements of J .
1. Let us firstly build three strategies: S
∗
1
, S
∗
2
, and S
∗
3
as follows.
(a) (S
∗
1
)
k
= (S
1
)
k
A
, (S
∗
2
)
k
= (S
2
)
k
A
, and (S
∗
3
)
k
=
(S
3
)
k
A
, if k 6 k
0
.
(b) Let us now explain how to replace X (X
B
)
j
q
,
∀q ∈ J0;λ − 1K:
First of all, we must replace X (X
B
)
j
0
:
CHAOTIC ITERATIONS FOR STEGANOGRAPHY - Stego-security and Chaos-security
225

i. If ∃λ
0
∈ J0;P − 1K/X (X
B
)
j
0
= ˇm
λ
0
, then we
can choose (S
∗
1
)
k
0
+1
= j
0
, (S
∗
2
)
k
0
+1
= λ
0
,
(S
∗
3
)
k
0
+1
= λ
0
, and so I
j
0
will be equal to 1.
ii. If such a λ
0
does not exist, we choose:
(S
∗
1
)
k
0
+1
= j
0
, (S
∗
2
)
k
0
+1
= 0, (S
∗
3
)
k
0
+1
= 0,
(S
∗
1
)
k
0
+2
= j
0
, (S
∗
2
)
k
0
+2
= 0, (S
∗
3
)
k
0
+2
= 0
and so let us notice I
j
0
= 2.
All of the X (X
B
)
j
q
are replaced similarly. The
other terms of S
∗
1
, S
∗
2
, and S
∗
3
are constructed
identically, and the values of I
j
q
are defined
on the same way.
Let γ =
∑
λ−1
q=0
I
j
q
.
(c) (S
∗
1
)
k
= (S
∗
1
)
j
, (S
∗
2
)
k
= (S
∗
2
)
j
and (S
∗
3
)
k
= (S
∗
3
)
j
where j 6 k
0
+ γ is satisfying j ≡ k [mod (k
0
+
γ)], if k > k
0
+ γ.
So,G
k
0
+γ
f
0
((S
∗
1
,x
A
,S
∗
2
,m
A
,S
∗
3
)) = (S
∗
1
,x
B
,S
∗
2
,m,S
∗
3
)
Let K =

i ∈ J0;P − 1K/m
i
6= M (X
B
)
i
, where
(S
∗
1
,x
B
,S
∗
2
,m,S
∗
3
) = G
k
0
+γ
f
0
((S
∗
1
,x
A
,S
∗
2
,m
A
,S
∗
3
))
o
,
µ = card(K ) and r
0
< r
1
< ... < r
µ−1
the elements
of K .
2. Let us secondly build three other strategies:
e
S
1
,
e
S
2
,
e
S
3
as follows.
(a)
e
S
1
k
= (S
∗
1
)
k
,
e
S
2
k
= (S
∗
2
)
k
, and
e
S
3
k
= (S
∗
3
)
k
, if
k 6 k
0
+ γ.
(b) Let us now explain how to replace
M (X
B
)
r
q
,∀q ∈ J0;µ − 1K:
First of all, we must replace M (X
B
)
r
0
:
i. If ∃µ
0
∈ J0;N − 1K/M (X
B
)
r
0
= (x
B
)
µ
0
, then
we can choose
e
S
1
k
0
+γ+1
= µ
0
,
e
S
2
k
0
+γ+1
= r
0
,
e
S
3
k
0
+γ+1
= r
0
, and J
r
0
will be equal to 1.
ii. If such a µ
0
does not exist, we choose:
e
S
1
k
0
+γ+1
= 0,
e
S
2
k
0
+γ+1
= r
0
,
e
S
3
k
0
+γ+1
= r
0
,
e
S
1
k
0
+γ+2
= 0,
e
S
2
k
0
+γ+2
= r
0
,
e
S
3
k
0
+γ+2
= 0,
e
S
1
k
0
+γ+3
= 0,
e
S
2
k
0
+γ+3
= r
0
,
e
S
3
k
0
+γ+3
= 0,
and so let us notice J
r
0
= 3.
All the M (X
B
)
r
q
are replaced similarly. The
other terms of
e
S
1
,
e
S
2
, and
e
S
3
are constructed
identically, and the values of J
r
q
are defined
on the same way.
Let α =
∑
µ−1
q=0
J
r
q
.
(c) ∀k ∈ N
∗
,
e
S
1
k
0
+γ+α+k
= (S
1
)
k
B
,
e
S
2
k
0
+γ+α+k
=
(S
2
)
k
B
, and
e
S
3
k
0
+γ+α+k
= (S
3
)
k
B
.
So, G
k
0
+γ+α
f
0
(
e
S
1
,x
A
,
e
S
2
,m
A
,
e
S
3
) = X
B
, with
(
e
S
1
,
e
S
2
,
e
S
3
) ∈ S
X
A
,k
0
. Then
e
X = (
e
S
1
,x
A
,
e
S
2
,m
A
,
e
S
3
) ∈
X
2
is such that
e
X ∈ B
A
and G
k
0
+γ+α
f
0
(
e
X) ∈ B
B
. Finally
we have proven the result.
8.3 Sensitivity on Initial Conditions
Proposition 9. (X
2
,G
f
0
) has sensitive dependence on
initial conditions.
Proof. G
f
0
is regular and transitive. Due to Theo-
rem 1, G
f
0
is sensitive.
8.4 Devaney’s Chaos
In conclusion, (X
2
,G
f
0
) is topologically transitive,
regular, and has sensitive dependence on initial con-
ditions. Then we have the result.
Theorem 3. G
f
0
is a chaotic map on (X
2
,d
2
) in the
sense of Devaney.
So we can claim that:
Theorem 4. SCISMM is chaos-secure.
9 CONCLUSIONS
In this research work, a new information hiding
scheme has been introduced. It is chaos-secure and
stego-secure, and thus is able to withstand attacks
in Watermark-Only Attack (WOA) and Constant-
Message Attack (CMA) setups. These results have
been obtained after having studied the topological be-
havior of this data hiding scheme. To the best of our
knowledge, this algorithm is the third scheme that has
been proven to be secure, according to the informa-
tion hiding security field.
In future work, we intend to study the robust-
ness of this scheme, and to compare it with the two
other secure algorithms. Additionally, we will inves-
tigate the topological properties of our scheme, to see
whether it is secure in KOA and KMA setups.
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