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%%%%%%%%%%%% ATTENTION : A VIRER POUR LA SOUMISSION 
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%\copyrightyear{2009} \vol{00} \issue{0} \DOI{000}

\begin{document}



%\title[Modelling Bidders in Sequential Automated Auctions]{Modelling Bidders in
%Sequential Automated Auctions}

\title[Information Hiding by Chaotic Iterations, Security and
 Robustness]{Information Hiding by Chaotic Iterations\\Security and Robustness}

%\author{Kumaara Velan}

% \author{Jacques M. Bahi\textsuperscript{1}, Nicolas Friot\textsuperscript{1},
% Christophe Guyeux\textsuperscript{1}, and Kamel
% Mazouzi\textsuperscript{2}~*\\\tiny * Authors in alphabetic order.}

\author{* Jacques M. Bahi}
\email{\{jacques.bahi,christophe.guyeux,kamel.mazouzi\}@univ-fcomte.fr\\nicolas.friot@lifc.univ-fcomte.fr\\\vspace{0.5cm}*
Authors are cited in alphabetic order.}

\author{Nicolas Friot}
\author{Christophe Guyeux}

\affiliation{Computer Science Laboratory LIFC}
\author{Kamel Mazouzi}
\affiliation{M\'esocentre de calcul de Franche-Comt\'e\\University of
Franche-Comt\'e, 16 route de Gray, Besan\c con, France\\
}

%\\\tiny Authors in alphabetic order.

% \affiliation{Intelligent Systems and Networks Group, Department of
% Electrical and Electronic Engineering, Imperial College, London
% SW7 2BT, UK} \email{kumaara.velan@imperial.ac.uk}

% \affiliation{\textsuperscript{1}Computer Science Laboratory LIFC\\
% %, University of Franche-Comt\'e\\
% %16 route de Gray, Besan\c con, France\\
% %\vspace{0.5cm}\\
% \textsuperscript{2}M\'esocentre de calcul de Franche-Comt\'e\\
% University of Franche-Comt\'e, 16 route de Gray, Besan\c con, France
% }

%\email{\{jacques.bahi,christophe.guyeux,kamel.mazouzi\}@univ-fcomte.fr\\nicolas.friot@lifc.univ-fcomte.fr}


%\shortauthors{K. Velan}
\shortauthors{J. M. Bahi, N. Friot, C. Guyeux, and K. Mazouzi}
% \shortauthors{N. Friot}
% \shortauthors{C. Guyeux}
% \shortauthors{K. Mazouzi}

% \received{00 January 2009}
% \revised{00 Month 2009}

\received{\today}
\revised{\today}

%\category{C.2}{Computer Communication Networks}{Computer Networks}
%\category{C.4}{Performance of Systems}{Analytical Models}
%\category{G.3}{Stochastic Processes}{Queueing Systems}
%\terms{Internet Technologies, E-Commerce}
% \keywords{Automated Auctions; Analytical Models; Autonomic
% Systems; Internet Technologies; E-Commerce; Queueing Systems}

\keywords{Information hiding; Chaos; Security; Robustness.
}



\begin{abstract}
In this paper, recent contributions in the field of chaos-based information hiding security is summed up and deepened.
Additionally, two variants of an information hiding scheme based on chaotic iterations are recalled and their security proofs are completed.
Among other things, new evaluations of qualitative and quantitative properties of security concerning the improved version of our algorithm are proposed.
The security notions used here encompass the probabilistic stego-security against watermark-only attacks, and the topological chaos-security to face known message, known original, and constant-message attacks.
Finally, a complete and original evaluation of the robustness of the first variant in spatial domain is given, encompassing several kinds of attacks against a set of hundreds of watermarked medias.
\end{abstract}

\maketitle


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\section{Introduction}\label{sec:introduction}


Information hiding has recently become a major security technology, especially with the increasing importance and widespread distribution of digital media through the Internet.
It includes several techniques, among which are digital watermarking and steganography. On the one hand, the aim of digital watermarking is to embed a piece of information into digital documents, like pictures or movies, for a large panel of reasons, such as: copyright protection, control utilization, data description, integrity checking, or content authentication. Digital watermarking must have an essential characteristic called robustness against attacks. On the other hand, steganography consisting in the deployment of an hidden channel in public exchanges, requires security properties guaranteeing that the channel is undetectable. 

Robustness and security are thus two major concerns in information hiding.
Even if security and robustness are neighboring concepts without clearly established definitions~\cite{Perez-Freire06}, robustness is often considered to be mostly concerned with blind elementary attacks, whereas security is not limited to certain specific attacks.
Indeed, security encompasses robustness and intentional attacks~\cite{Kalker2001,ComesanaPP05bis}. 
The best attempt to give an elegant and concise definition for each of these two terms was proposed in \cite{Kalker2001}.
Following Kalker, we will consider in this research work that:
 ``Robust
watermarking is a mechanism to create a communication channel that is
multiplexed into original content [...]. It is required that, firstly, the
perceptual degradation of the marked content [...] is minimal and, secondly,
that the capacity of the watermark channel degrades as a smooth function of the
degradation of the marked content. [...]. Watermarking security 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.''


In the framework of watermarking and steganography, security has seen several important developments since the last decade~\cite{BarniBF03,Cayre2005,Ker06}. 
The first fundamental work in security was made by Cachin in the context of steganography~\cite{Cachin2004}. 
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 stegosystem are defined using the notions of entropy, mutual information, and relative entropy. 
Mittelholzer, inspired by the work of Cachin, proposed the first theoretical framework for analyzing the security of a watermarking scheme~\cite{Mittelholzer99}.
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 versus security), and the classifications of watermarking attacks~\cite{Kalker2001}. 
This work has been deepened by Furon \emph{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~\cite{Furon2002}. 
They used Diffie and Hellman methodology, and Shannon's cryptographic framework~\cite{Shannon49}, to classify the watermarking attacks into categories, according to the type of information Eve has access to~\cite{Cayre2005,Perez06}, namely: Watermarked Only Attack (WOA), Known Message Attack (KMA), Known Original Attack (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 \cite{Cayre2008}, whereas chaos-security tends to improve the ability to withstand attacks in KMA, KOA, and CMA setups \cite{gfb10:ip}.


This paper is both an overview of our contributions in the field of information hiding~\cite{gfb10:ip,guyeuxThese10,guyeux10ter}, an extension of our researches concerning the security of chaotic iterations based schemes, and a complete study of their robustness. In \cite{gfb10:ip,guyeuxThese10}, we have shown that the security level of such algorithms can be studied into a novel framework based on unpredictability, as it is understood in the mathematical theory of chaos~\cite{Devaney}. To do so, a new class of security has been introduced, namely the chaos-security. This new class can be used to study some categories of attacks that are difficult to investigate in the existing security approach: namely, the KMA, KOA, and CMA setups. It also enriches the variety of qualitative and quantitative tools that evaluate how strong the security is, thus reinforcing the confidence that can be had in a given scheme. 

We have applied this framework in concrete examples. For instance, we have proven in~\cite{guyeuxThese10,ih10} that Natural Watermarking (NW) technique is chaos-secure. Moreover, this technique possesses additional properties of unpredictability, namely, strong transitivity, topological mixing, and a constant of sensitivity equal to $\frac{N}{2}$. However NW are not expansive, which is problematic in the CMA and KMA setups~\cite{guyeuxThese10,ih10}. 
This is why a new information hiding scheme has been introduced in~\cite{guyeux10ter}, which is based on the so-called chaotic iterations. Its security has been established in \cite{gfb10:ip}. However, this former algorithm allow to embed only one bit. So this scheme has been improved in \cite{fgb11:ip}, in a version that allows to embed $n$ bits, while preserving the highest levels of security. 

In this research work, all of these contributions are regrouped together. This summary is the first contribution of this paper. The security study of the new version of our information hiding scheme is deepened, by giving qualitative and quantitative measures of its level of chaos-security. This second contribution encompasses the evaluation of the constant of sensibility, and the qualitative properties of strong transitivity and indecomposability. Finally, a complete and rigorous evaluation of the robustness of the algorithm is proposed as a third contribution. This study has been initiated in \cite{guyeux10ter}, but only on one given image and for a small number of attacks. We have extend this work by using hundreds of images and a large variety of parameters and attacks.

The remainder of this document is organized as follows.
In Section \ref{sec:basic-recalls}, some basic recalls concerning both chaotic iterations and Devaney's chaos are given, and the relation unifying them is recalled.
In Section \ref{section:IH based on CIs} is presented the first version, denoted by $CI_1$, of our information hiding scheme.
The next section is devoted to the security study of $CI_1$, encompassing both the stego-security and the chaos-security.
Then, in Section \ref{section:robustness-ci-1}, a complete and rigorous evaluation of the robustness of $CI_1$, realized by using the ``M\'{e}socentre de calcul de Franche-Comt\'{e}'', is given.

This research work ends by a conclusion section where our contribution is
summarized and intended future researches are presented.





\section{Basic Recalls}
\label{sec:basic-recalls}


\subsection{Chaotic Iterations}
\label{sec: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: $\llbracket0;N\rrbracket=\{0,1,\hdots,N\}$.\newline

Let us consider a \emph{system} of a finite number $\mathsf{N}$ of elements (or
\emph{cells}), so that each cell has a boolean \emph{state}. A sequence of length
$\mathsf{N}$ of boolean states of the cells corresponds to a particular
\emph{state of the system}. A sequence that elements belong into $\llbracket
0;\mathsf{N-1} \rrbracket $, \emph{i.e.}, an element of $\llbracket 0; \mathsf{N-1}\rrbracket^\mathds{N}$, is called a \emph{strategy}. The set of all
strategies is denoted by $\mathbb{S}.$

\begin{definition}
\label{Def:chaotic iterations}
The set $\mathds{B}$ denoting $\{0,1\}$, let
$f:\mathds{B}^{\mathsf{N}}\longrightarrow \mathds{B}^{\mathsf{N}}$ be a function
and $S\in \mathbb{S}$ be a strategy.
The so-called \emph{chaotic iterations} are
defined by $x^0\in \mathds{B}^{\mathsf{N}}$ and $\forall (n,i) \in
\mathds{N}^{\ast} \times \llbracket0;\mathsf{N-1}\rrbracket$:
\begin{equation*}
%\forall n\in \mathds{N}^{\ast }, \forall i\in \llbracket1;\mathsf{N}\rrbracket
%,x_i^n=\left\{
x_i^n=\left\{
\begin{array}{ll}
x_i^{n-1} & \text{ if }S^n\neq i, \\
\left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.\end{array}\right.\end{equation*}
\end{definition}


In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
\textquotedblleft iterated\textquotedblright . Note that in a more general
formulation, $S^n$ can be a subset of components and $f(x^{n-1})_{S^{n}}$ can
be replaced by $f(x^{k})_{S^{n}}$, where $k<n$, describing for example, delays
transmission (see \cite{Bahi2002,bcv06:ij}). For the
general definition of such chaotic iterations, see,
\emph{e.g.},~\cite{Robert1986}.



\subsection{Devaney's chaotic dynamical systems}
\label{sec:Devaney}

Some topological definitions and properties taken from the
mathematical theory of chaos are recalled in this section.
%It is important to understand these notions which will be used in
%the rest of this article especially in the Section~\ref{sec:SCISMM-topological-model}~\vpageref{sec:SCISMM-topological-model} and the Section~\ref{sec:SCISMM-chaos-security}~\vpageref{sec:SCISMM-chaos-security}.

%\subsubsection{Notations}

%\begin{itemize}
%  \item Let $(\mathcal{X},d)$ be a metric space.
%  \item Let $f$ be a continuous function on  $(\mathcal{X},d)$.
%\end{itemize}

Let $(\mathcal{X},d)$ be a metric space and $f$ a continuous function on  $(\mathcal{X},d)$.

%\subsubsection{Topological definitions used in chaos}

\begin{definition}[Topological transitivity]
 $f$ is said to be \emph{topologically transitive} if, for any pair
of open sets $U,V\subset \mathcal{X}$, there exists $k>0$ such that $f^{k}(U)\cap
V\neq\varnothing $.
\label{def:chaos-devaney}
\end{definition}

\begin{definition}[Regularity]
$(\mathcal{X},f)$ is said to be \emph{regular} if the set of
periodic points is dense in $\mathcal{X}$.
\end{definition}

\begin{definition}[Sensitivity]
$f$ has \emph{sensitive dependence on initial conditions} if there exists $\delta >0$ such that, for any $x\in
\mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and
$n\geqslant 0$ such that $d\left(f^{n}(x),f^{n}(y)\right)>\delta $.

$\delta $ is called the \emph{constant of sensitivity} of $f$.
\end{definition}


It is now possible to introduce the well-established mathematical definition of
chaos formulated by Devaney~\cite{Devaney89},

\begin{definition}[Chaotic function]
A function $f:\mathcal{X}\longrightarrow \mathcal{X}$ is said to be
\emph{chaotic} on $\mathcal{X}$ if:
\begin{enumerate}
  \item $f$ is regular,
  \item $f$ is topologically transitive,
  \item $f$ has sensitive dependence on initial conditions.
\end{enumerate}
\end{definition}

When $f$ is chaotic, then the system $(\mathcal{X}, f)$ is chaotic and quoting
Devaney: ``it is unpredictable because of the sensitive dependence on initial
conditions. It cannot be broken down or simplified into two subsystems which do
not interact because of topological transitivity. And in the midst of this
random behavior, we nevertheless have an element of regularity''. Fundamentally
different behaviors are consequently possible and occur in an unpredictable
way.

Let us finally remark that,


\begin{theorem}[Banks~\cite{Banks92}]\label{theo:banks}
If a function is regular and topologically transitive on a metric space, then the
function is sensitive on initial conditions.
\end{theorem}

%The main interest of the Banks theorem is to reduce the properties to check when
%we want to prove that a function on a metric space is chaotic. 
%This theorm will be used in the
%Section~\ref{sec:SCISMM-chaos-security}~\vpageref{sec:SCISMM-chaos-security}.

%This theorem reduces the number of proofs when establishing the chaotic behavior of an iterate function on a metric space.
%We will use it, for instance, in Section~\ref{sec:SCISMM-chaos-security}.

Let us now study the original relation between chaotic iterations and Devaney's chaos. This relation will be used in the context of information hiding security. 

\subsection{Dynamics of Chaotic Iterations}\label{sec:topological}


\noindent In this section, we give the outline proofs we have previously obtained, establishing the topological properties of chaotic iterations \cite{gfb10:ip,guyeux10ter,bg10:ij}. As our scheme is inspired by this work, the proofs detailed in this document will follow a same canvas.

Let us firstly introduce some notations and terminologies in a view to describe chaotic iterations as an iteration function on a metric space. First of all, we show that CIs consist in the operation of a specific function $F_f$ on some components of a system. This can be described by using a discrete Boolean metric, as follows.



\begin{definition}[Discrete boolean metric]
\label{def:discret-boolean-metric}
The \emph{discrete boolean metric} is the
application $\delta: \mathds{B} \longrightarrow \mathds{B}$ defined by
$\delta(x,y)=0\Leftrightarrow x=y.$
\end{definition}

We can now define $F_f$, whose goal is to change the $k-$th component of the state $E$ by using $f$:

\begin{definition}[Function $F_{f}$]
Given a function $f:\mathds{B}^{\mathsf{N}} \rightarrow \mathds{B}^{\mathsf{N}}$, the function \emph{$F_{f}$} is defined by:
\begin{equation*}
\begin{array}{ll}
F_{f}: & \llbracket 0;\mathsf{N-1}\rrbracket \times
\mathds{B}^{\mathsf{N}} \longrightarrow \mathds{B}^{\mathsf{N}} \\
 & (k,E) \longmapsto \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
(k,j)}\right)_{j\in \llbracket 0;\mathsf{N-1}\rrbracket} \\
\end{array}
\end{equation*}
\end{definition}

As recalled before, the components to update are given by a ``strategy'', \emph{i.e.}, a sequence of integers bounded by the size of the system. This can be formulated as follows,

\begin{definition}[Strategy]
\label{def:strategy-adapter}
Let $\mathsf{k} \in \mathds{N}^\ast$. 
A \emph{strategy} is a sequence which elements belong into $\llbracket 0, \mathsf{k-1} \rrbracket$.
The set of all strategies with terms in $\llbracket 0, \mathsf{k-1} \rrbracket$
is denoted by  $\mathbb{S}_\mathsf{k}$.
\end{definition}

We thus iterate on the Cartesian product $\mathcal{X}_1$ constituted by the set of states and the set of strategies:

\begin{definition}[Phase space]
The phase space used for chaotic iterations is denoted by $\mathcal{X}_1$ and
defined by $\mathcal{X}_1=\mathbb{S}_\mathsf{N} \times
\mathds{B}^{\mathsf{N}}.$
\end{definition}

We can remark that~\cite{bg10:ij},

\begin{proposition}%[Cardinality of $\mathcal{X}_1$]
\label{prop:cardinality-X1}
The phase space $\mathcal{X}_1$ has, at least, the cardinality of the 
continuum.
\end{proposition}



It still remains to describe CIs as a discrete dynamical system $x^{n+1} = G_f(x^n)$ on $\mathcal{X}_1$ for a function $G_f$ to define.
To do so, we will use $F_f$ and two other functions, namely the \emph{initial} and the \emph{shift} functions.
The initial function returns the first term of a given strategy-adapter whereas the shift function removes it:

\begin{definition}[Initial function]
Let $\mathsf{k} \in \mathds{N}^\ast$. 
The \emph{initial function} is the map $i_\mathsf{k}$ defined by:
\begin{equation*}
\begin{array}{cccc}
i_\mathsf{k}: & \mathbb{S}_{\mathsf{k}} & \longrightarrow & \llbracket 0, \mathsf{k-1} \rrbracket \\
   & (S^{n})_{n\in \mathds{N}} & \longmapsto & S^{0} \\
\end{array}
\end{equation*}
\end{definition}




\begin{definition}[Shift function]
Let $\mathsf{k} \in \mathds{N}^\ast$. 
The \emph{shift function} is the map $\sigma_\mathsf{k}$ defined by:
\begin{equation*}
\begin{array}{cccc}
\sigma_\mathsf{k} : & \mathbb{S}_{\mathsf{k}} & \longrightarrow & \mathbb{S}_\mathsf{k}\\
         & (S^{n})_{n\in \mathds{N}} & \longmapsto & (S^{n+1})_{n\in \mathds{N}}
\end{array}
\end{equation*}
\end{definition}

We are now able to introduce the function $G_f$:

\begin{definition}[Map $G_{f}$]
Given a function $f:\mathds{B}^{\mathsf{N}} \rightarrow \mathds{B}^{\mathsf{N}}$, the map \emph{$G_{f}$} is defined by:
\begin{equation*}
\begin{array}{cccc}
G_{f}: & \mathcal{X}_1 & \longrightarrow & \mathcal{X}_1 \\
 & \left( S,E\right) & \longmapsto &\left( \sigma_\mathsf{N} (S),F_{f}(i_\mathsf{N}(S),E)\right)
 \\
\end{array}
\end{equation*}
\end{definition}





In other words, one iteration consists in a shift of strategy $S$ and a replacement of the $i_\mathsf{N}(S)$ cell with $f(E)_{i_\mathsf{N}(S)}$.


With these definitions, CIs can be described by the following iterations of the discrete dynamical system:

\begin{equation}
\left\{
\begin{array}{l}
X^{0}\in \mathcal{X}_1\\
\forall k \in \mathds{N}^\ast, X^{k+1}=G_{f}(X^{k})
\end{array}
\right.
\end{equation}


We need a metric space in order to study the topological behavior of the CIs. This is why a new distance $d_1$ between two points has been defined in~\cite{bg10:ij} by:


\begin{definition}[Distance $d_1$ on  $\mathcal{X}_1$]\label{def:distance-on-X1}
$\forall (S,E),(\check{S},\check{E} )\in \mathcal{X}_1,$
$d_1((S,E);(\check{S},\check{E}))=d_{\mathds{B}^{\mathsf{N}}}(E,\check{E})+d_{\mathbb{S}_\mathsf{N}}(S,\check{S}),$
where:
\begin{itemize}
\item
$\displaystyle{d_{\mathds{B}^{\mathsf{N}}}(E,\check{E})}=\displaystyle{\sum_{k=0}^{\mathsf{N-1}}\delta
(E_{k},\check{E}_{k})} \in \llbracket 0 ; \mathsf{N} \rrbracket$
\item
$\displaystyle{d_{\mathbb{S}_\mathsf{N}}(S,\check{S})}=\displaystyle{\dfrac{9}{\mathsf{N}}\sum_{k=0}^{\infty
}\dfrac{|S^{k}-\check{S}^{k}|}{10^{k+1}}} \in [0 ; 1[.$
\end{itemize}
are respectively two distances on $\mathds{B}^{\mathsf{N}}$ and $\mathbb{S}_\mathsf{N}$
($\forall \mathsf{N} \in \mathds{N}^\ast$).
\end{definition}

\begin{remark}
This new distance has been introduced in \cite{bg10:ij}
 to satisfy the following
requirements. When the number of different 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. 
Indeed, if the floor value $\lfloor
d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ differ in $n$
cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a measure of the
differences between strategies $S$ and $\check{S}$. More precisely, this floating
part is less than $10^{-k}$ if and only if the first $k$ terms of the two
strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the
$k^{th}$ terms of the two strategies are different.
\end{remark}



We have proven that,

\begin{proposition}\label{Prop:continuite}
$G_f$ is a continuous function on $(\mathcal{X}_1,d_1)$, for all $f:\mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$.
\end{proposition}

Let us finally recall the iteration function used in \cite{guyeux10ter}.


\begin{definition}%[Vectorial negation]
The \emph{vectorial negation} is the function defined by:
\begin{equation*}
\begin{array}{cccc}
f_{0} : & \mathbb{B}^\mathsf{N} & \longrightarrow & \mathbb{B}^\mathsf{N}\\
& (b_0,\cdots,b_\mathsf{N-1}) & \longmapsto &
(\overline{b_0},\cdots,\overline{b_\mathsf{N-1}})\\
\end{array}
\end{equation*}
\end{definition}
%\begin{definition}[Vectorial Negation]\label{Def:vectorial-negation}
%\begin{equation}
%\begin{array}{cccc}
%f_0\ :\ & \mathbb{B}^N & \longrightarrow & \mathbb{B}^N \\
%& (b_1,\cdots,b_\mathsf{N}) & \longmapsto &
%(\overline{b_1},\cdots,\overline{b_\mathsf{N}})\\
%\end{array}
%\end{equation}
%\end{definition}
%It is then checked in \cite{bg10:ij} that i
In the metric space $(\mathcal{X}_1,d_1)$, $G_{f_0}$  satisfies the three
conditions for Devaney's chaos: regularity, transitivity, and
sensibility to the initial condition. %~\cite{bg10:ij}. 
So,

%the following result.
\begin{theorem}%[$G_{f_0}$ is a chaotic map]
$G_{f_0}$ is a chaotic map on $(\mathcal{X}_1,d_1)$ according to the Devaney's formulation.
\end{theorem}


\section{A First CIs Watermarking Process called $CI_1$}
\label{section:IH based on CIs}



In this section is explained how CIs can be used as an information hiding scheme.

\subsection{Most and Least Significant Coefficients}\label{sec:msc-lsc}



We first take into account the fact that, in a watermarking process, 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 
\emph{signification function} 
attaches a weight to each term defining a digital media,
depending on its position $t$ \cite{todo}.

\begin{definition}[Signification function]
A \emph{signification function} is a real sequence 
$(u^k)^{k \in \mathds{N}}$. % with a limit equal to 0.
\end{definition}


\begin{example}\label{Exemple LSC}
Let us consider a set of    
grayscale images stored into portable graymap format (P3-PGM):
each pixel ranges between 256 gray levels, \textit{i.e.},
is memorized with eight bits.
In that context, we consider 
$u^k = 8 - (k  \mod  8)$  to be the $k$-th term of a signification function 
$(u^k)^{k \in \mathds{N}}$. 
Intuitively, in each group of eight bits (\textit{i.e.}, for each pixel) 
the first bit has an importance equal to 8, whereas the last bit has an
importance equal to 1. This is compliant with the idea that
changing the first bit affects more the image than changing the last one.
\end{example}

\begin{definition}[Significance of coefficients]
\label{def:msc,lsc}
Let $(u^k)^{k \in \mathds{N}}$ be a signification function, 
$m$ and $M$ be two reals s.t. $m < M$. 
\begin{itemize}
\item The \emph{most significant coefficients (MSCs)} of $x$ is the finite 
  vector  $$u_M = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k 
    \geqslant M \textrm{ and }  k \le \mid x \mid \right);$$
 \item The \emph{least significant coefficients (LSCs)} of $x$ is the 
finite vector 
$$u_m = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k 
  \le m \textrm{ and }  k \le \mid x \mid \right);$$
 \item The \emph{passive coefficients} of $x$ is the finite vector 
   $$u_p = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } 
u^k \in ]m;M[ \textrm{ and }  k \le \mid x \mid \right).$$
 \end{itemize}
 \end{definition}

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~\ref{fig:MSCLSC}, where the significance function $(u^k)$ is defined as in Example \ref{Exemple LSC}, $M=5$, and $m=6$.

\begin{figure*}[htb]
\begin{center}

\begin{minipage}[b]{.98\linewidth}
  \centering
  \centerline{\includegraphics[width=6.cm]{img/lena512}}
   %\centerline{\epsfig{figure=img/lena512.pdf,width=4cm}}
  \centerline{(a) Original Lena.}
\end{minipage}
\begin{minipage}[b]{.49\linewidth}
  \centering
    \centerline{\includegraphics[width=6.cm]{img/lena_msb_678}}
   %\centerline{\epsfig{figure=img/lena_msb_678.pdf,width=4cm}}
  \centerline{(b) MSCs of Lena.}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\linewidth}
  \centering
    \centerline{\includegraphics[width=6.cm]{img/lena_lsb_1234_facteur17}}
  
 %\centerline{\epsfig{figure=img/lena_lsb_1234_facteur17.pdf,width=4cm}}
  \centerline{(c) LSCs of Lena ($\times 17$).}
\end{minipage}
%
\caption{Most and least significant coefficients of Lena.}
\label{fig:MSCLSC}
\end{center}

\end{figure*}










\subsection{Presentation of the Scheme}

We have proposed in \cite{guyeux10ter} to use chaotic iterations as an information hiding scheme, as follows. 
Let:
\begin{itemize}
  \item $(K,N) \in [0;1]\times \mathds{N}$ be an embedding key,
  \item $X \in \mathbb{B}^\mathsf{N}$ be the $\mathsf{N}$ LSCs of a cover $C$,% $X$ be the initial state $X_0$,
  \item $(S^n)_{n \in \mathds{N}} \in \llbracket 0, \mathsf{N-1}
  \rrbracket^{\mathds{N}}$ be a strategy, which depends on the message to hide $M \in [0;1]$ and $K$,
  \item $f_0 : \mathbb{B}^\mathsf{N} \rightarrow \mathbb{B}^\mathsf{N}$ be the vectorial logical negation.
\end{itemize}


So the watermarked media is $C$ whose LSCs are replaced by $Y_K=X^{N}$, where:

\begin{equation*}
\left\{
 \begin{array}{l}
X^0 = X\\
\forall n < N, X^{n+1} = G_{f_0}\left(X^n\right).\\
\end{array} \right.
\end{equation*}


Two ways to generate $(S^n)_{n \in \mathds{N}}$ are given in \cite{trx}, namely
Chaotic Iterations with Independent Strategy~(CIIS) and Chaotic Iterations with Dependent
Strategy~(CIDS). 
In CIIS, the strategy is independent from the cover media $C$, whereas in CIDS the strategy will be dependent on $C$. 



\subsection{CIIS Strategy}
\label{CIIS strategy}
Let us firstly give the definition of the Piecewise Linear Chaotic Map~(PLCM, see~\cite{Shujun1}):

%\begin{definition}%[PLCM]
%The \emph{Piecewise Linear Chaotic Map} is defined by
\begin{equation*}
F(x,p)=\left\{
 \begin{array}{ccc}
x/p & \text{if} & x \in [0;p], \\
(x-p)/(\frac{1}{2} - p) & \text{if} & x \in \left[ p; \frac{1}{2} \right],
\\
F(1-x,p) & \text{else,} & \\
\end{array} \right.
\end{equation*}
%\end{definition}


\noindent where $p \in \left] 0; \frac{1}{2} \right[$ is a ``control parameter''.

The general term of the strategy $(S^n)_n$ in CIIS setup is defined by
the following expression: $S^n = \left \lfloor \mathsf{N} \times K^n \right \rfloor +
1$, where:

\begin{equation*}
\left\{
 \begin{array}{l}
p \in \left[ 0 ; \frac{1}{2} \right] \\
K^0 = M \otimes K\\
K^{n+1} = F(K^n,p), \forall n \leq N_0, \end{array} \right.
\end{equation*}



\noindent in which $\otimes$ denotes the bitwise exclusive or (XOR) between two floating part numbers (\emph{i.e.}, between their binary digits representations). 


\subsection{CIDS Strategy}

The same notations as above are used.
We define CIDS strategy as follows: $\forall k \leqslant N$,  
\begin{itemize}
\item if $k \leqslant \mathsf{N}$ and $X^k = 1$, then $S^k=k$,
\item else $S^k=1$.
\end{itemize}
In this situation, if $N \geqslant \mathsf{N}$, then only two watermarked contents are possible with the information hiding scheme proposed previously, namely: $Y_K=(0,0,\cdots,0)$ and $Y_K=(1,0,\cdots,0)$.


\subsection{PSNR evaluation}\label{section:psnr-ci-1}

To realize the evaluation of the PSNR of $CI_1$, we have used the same
architecture as described in
Section~\ref{section:architecture-presentation}~\vpageref{section:architecture-presentation}.

It has to be noted that in the $CI_1$ watermarking process, the embedding key
correspond to the strategy associated to the number of iterations.  

Executing the $CI_1$ process on 1000 images, on the one hand with a  constant
embedding key and one the other hand with a key generated randomly we have
obtain the results described in the
table~\ref{table:psnr-ci1}~\vpageref{table:psnr-ci1}, an average of and a standard deviation of  has been obtained for the PSNR.\newline



\begin{table*}
\begin{center}

 \begin{tabular}{|c||c|c|c|}
  \hline
   \textbf{Strategies} &  \textbf{PSNR: average} &  \textbf{PSNR: standard
   deviation}\\
    \hline 
    \hline 
 \textbf{Constant strategy} &  $21.6653$ &  $0.03604$  \\
  \hline
 \textbf{Random strategy}  & $13.3909$ & $10.5690$  \\
  \hline
 
  \hline 
  \end{tabular}
  \caption{PSNR evaluation of the $CI_1$ process (tests on 1000 images).}
  \end{center}
  \label{table:psnr-ci1}
  \end{table*}

Let us now focus on some security aspects of information hiding schemes. In particular, we aim to explain why chaos is relevant in this field.

\section{Information hiding security: $CI_1$}
\label{section:IH-security}

\subsection{Classification of Attacks}\label{sec:attack-classes}

In the steganography framework, attacks have been classified
in~\cite{Cayre2008} as follows. 


\begin{definition}
Watermark-Only Attack (WOA) occurs when an attacker has only access to several watermarked contents.
\end{definition}

\begin{definition}
Known-Message Attack (KMA) occurs when an attacker has access to several pairs of watermarked contents and
corresponding hidden messages.
\end{definition}

\begin{definition}
Known-Original Attack (KOA) is when an attacker has access to several pairs of watermarked contents and
their corresponding original versions.
\end{definition}

\begin{definition}
Constant-Message Attack (CMA) occurs when the attacker observes several watermarked contents and only knows that the
unknown hidden message is the same in all contents.
\end{definition}

The synthesis of this classification is illustrated in the
Table~\ref{table:attack-classification}~\vpageref{table:attack-classification}.


\begin{table*}
\begin{center}

 \begin{tabular}{|c||c|c|c|}
  \hline
   \textbf{Attacks} &  \textbf{Original content} &  \textbf{Stego
  content} &  \textbf{Hidden message}\\ 
    \hline 
    \hline 
 \textbf{Watermark-Only Attack} &   &    $\times$ & \\
  \hline
 \textbf{Known-Message Attack}  & & $\times$ & $\times$ \\
  \hline
 \textbf{ Kown-Original Attack} & $\times$ & $\times$ & \\
 \hline
 \textbf{Constant-Message Attack} & & & $\times$ \\
  \hline 
  \end{tabular}
  \caption{Attacks in Kalker's context~\cite{Kalker2001}}
  \end{center}
  \label{table:attack-classification}
  \end{table*}
  

\subsection{Stego-Security of $CI_1$}\label{sec:stego-security}

In the prisoner problem of Simmons~\cite{Simmons83,DBLP:conf/ih/BergmairK06}
illustrated by the
Figure~\ref{fig:simmons-prisonner-problem}~\vpageref{fig:simmons-prisonner-problem}, Alice and Bob are in jail, and they want to, possibly, devise an escape plan by exchanging hidden messages in innocent-looking cover contents. These messages are to be conveyed 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.

\begin{figure*}
\begin{center}
%\includegraphics[width=10.5cm]{./img/prisoner-problem.png}
\includegraphics[width=15cm]{./img/prisoner-problem}
\end{center}
\caption{Simmons' prisoner problem~\cite{Simmons83}}
\label{fig:simmons-prisonner-problem}
\end{figure*}

The stego-security, defined in this
framework, is the highest security level in WOA setup~\cite{Cayre2008}.
To recall it, we need the following notations:
\begin{itemize}
  \item $\mathds{K}$ is the set of embedding keys,
  \item $p(X)$ is the probabilistic model of $N_0$ initial host contents,
  \item $p(Y|K_1)$ is the probabilistic model of $N_0$ watermarked contents.
\end{itemize}

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:

\begin{definition}[Stego-Security]
\label{Def:Stego-security}
The embedding function $e$ is \emph{stego-secure} if and only if:
$$\forall K_1 \in \mathds{K}, p(Y|K_1)=p(X).$$
\end{definition}

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 \cite{Cayre2008} have established that the spread spectrum technique called Natural Watermarking is stego-secure when its distortion parameter $\eta$ is equal to $1$.
On the other hand, it has been proven in~\cite{gfb10:ip} that: 

\begin{proposition}
Chaotic Iterations with Independent Strategy (CIIS)  are stego-secure.
\label{prop:CIIS-stego-security}
\end{proposition}



\begin{proof}
Let us suppose that $X \sim
\mathbf{U}\left(\mathbb{B}^N\right)$ in a CIIS setup. 
We will prove by a mathematical induction that $\forall n \in \mathds{N}, X^n \sim
\mathbf{U}\left(\mathbb{B}^N\right)$. The base case is immediate, as $X^0 = X \sim
\mathbf{U}\left(\mathbb{B}^N\right)$. Let us now suppose that the statement
 $X^n \sim
\mathbf{U}\left(\mathbb{B}^N\right)$ holds for some $n$. 
Let $e \in \mathbb{B}^N$ and $\mathbf{B}_k=(0,\cdots,0,1,0,\cdots,0) \in \mathbb{B}^N$ (the digit $1$ is in position $k$).
%\begin{equation*}
So $P\left(X^{n+1}=e\right)=\sum_{k=1}^N
P\left(X^n=e+\mathbf{B}_k,S^n=k\right).$
%\end{equation*}
These two events are independent in CIIS setup, thus:
%\begin{equation*}
$P\left(X^{n+1}=e\right)=\sum_{k=1}^N
P\left(X^n=e+\mathbf{B}_k\right) \times P\left(S^n=k\right)$.
%\end{equation*}
%According to the recurrence hypothesis ($X^n \sim
%\mathbf{U}\left(\mathbb{B}^N\right)$):
According to the inductive hypothesis:
%\begin{equation*}
%\begin{array}{ll}
%P\left(X^{n+1}=e\right) & =\sum_{k=1}^N
%\frac{1}{2^N} \times P\left(S^n=k\right) \\
%& =\frac{1}{2^N} \sum_{k=1}^N
% P\left(S^n=k\right)\\
%\end{array}
%\end{equation*}
%\begin{equation*}
$P\left(X^{n+1}=e\right)=\frac{1}{2^N} \sum_{k=1}^N
 P\left(S^n=k\right)$.
%\end{equation*}
The set of events $\left \{ S^n=k \right \}$ for $k \in \llbracket 1;N
\rrbracket$ is a partition of the universe of possible, so
$\sum_{k=1}^N P\left(S^n=k\right)=1$.

%Evaluate now $ P\left(S^n=k\right)$.
%\newline
%We have: $K^0 = M \otimes K \sim
%\mathbf{U}\left([0;1]\right)$, so $K^\prime=F\left( K^0,p \right) \sim
%\mathbf{U}\left([0;1]\right)$, because for PLCMs, ``Uniform input generate
%uniform output''~\cite{Lasota}.
%Well with an immediate proof by recurrence we have:
%$K^n \sim \mathbf{U}\left([0;1]\right)$. As $S^n = \left \lfloor N \times X^n
%\right \rfloor + 1$ we can deduce that: $S^n \sim
%\mathbf{U}\left([1;N]\right)$. Therefore: $ P\left(S^n=k\right)=\frac{1}{N}$.
Finally, 
%\begin{equation*}
%\begin{array}{llr}
%P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
% P\left(S^n=k\right) & \\
% & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} & \\
% & =\frac{1}{2^N} & \\
%\end{array}
%\end{equation*}
%\begin{equation*}
%$\begin{array}{llr}
%P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
% P\left(S^n=k\right) & \\
%P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} &
% =\frac{1}{2^N} \\
%\end{array}$.
$P\left(X^{n+1}=e\right)=\frac{1}{2^N}$, which leads to $X^{n+1} \sim
\mathbf{U}\left(\mathbb{B}^N\right)$.
This result is true $\forall n \in \mathds{N}$, we thus have proven that, $$\forall K \in [0;1], Y_K=X^{N_0} \sim
\mathbf{U}\left(\mathbb{B}^N\right) \text{ when } X \sim
\mathbf{U}\left(\mathbb{B}^N\right)$$

%already corrected
So CIIS defined in
Section~\ref{CIIS strategy} are stego-secure.
\end{proof}

We will now prove that,

\begin{proposition}
%already corrected
Chaotic Iterations with Dependent Strategy (CIDS)  are not stego-secure.
\end{proposition}

\begin{proof}
Due to the definition of CIDS, we have $P(Y_K=(1,1,\cdots,1))=0$. So there is
no uniform repartition for the stego-contents $Y_K$.
\end{proof}







\subsection{Chaos-Security of the $CI_1$ scheme}\label{sec: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
$(\mathcal{X},d)$. This formulation is always possible~\cite{ih10}. So,


\begin{definition}[Chaos-Security]
\label{Def:chaos-security-definition}
An information hiding scheme $S$ is said to be chaos-secure on
$(\mathcal{X},d)$ if its iterative process has a chaotic
behavior according to Devaney.
\end{definition}

In the approach presented by Guyeux \emph{et al.}, a data hiding scheme is secure if it is unpredictable. 
Its iterative process must satisfy the Devaney's chaos property 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~\cite{ih10} as a complementary approach to the existing framework. 
It contributes to the reinforcement of confidence into existing secure data hiding schemes. 
Additionally, the study of security in KMA, KOA, and CMA setups is realizable in this context. 
Finally, this framework can replace stego-security in situations that are not encompassed by it. 
In particular, this framework is more relevant to give evaluation of data hiding schemes claimed as chaotic. 




We can establish that,

\begin{proposition}
%already corrected
CIIS and CIDS are chaos-secure.
\end{proposition}

\begin{proof}
It is due to the fact that chaotic iterations have a chaotic behavior, as defined by Devaney. 
\end{proof}



In the two following sections, we will study the qualitative and quantitative
properties of chaos-security for chaotic iterations. These properties can measure the
disorder generated by our scheme, giving by doing so some important informations about the
unpredictability level of such a process.

\subsection{Quantitative property of
chaotic iterations}\label{sec:chaos-security-quantitative}

\begin{definition}[Expansivity]
A function $f$ is said to be \emph{expansive} if $
\exists \varepsilon >0,\forall x\neq y,\exists n\in \mathds{N}%
,d(f^{n}(x),f^{n}(y))\geqslant \varepsilon .$
\end{definition}

\begin{proposition}
 $G_{f_{0}}$ is an expansive chaotic dynamical system on $\mathcal{X}$ with a constant of expansivity is equal to 1.
\end{proposition}
\begin{proof}
If $(S,E)\neq (\check{S};\check{E})$, then either $E\neq \check{E}$, so at least
one cell is not in the same state in $E$ and $\check{E}$. Consequently the distance between $(S,E)$ and $(%
\check{S};\check{E})$ is greater or equal to 1. Or $E=\check{E}$. So the
strategies $S$ and $\check{S}$ are not equal. Let $n_{0}$ be the first index in which the terms $S$ and $\check{S}$
differ. Then $
\forall
k<n_{0},\tilde{G}_{f_{0}}^{k}(S,E)=\tilde{G}_{f_{0}}^{k}(\check{S},\check{E})$,
and $\tilde{G}_{f_{0}}^{n_{0}}(S,E)\neq \tilde{G}_{f_{0}}^{n_{0}}(\check{S},\check{E})$. As $E=\check{E},$ the cell which has changed in $E$ at the $n_{0}$-th
iterate is not the same as the cell which has changed in $\check{E}$, so
the distance between $\tilde{G}_{f_{0}}^{n_{0}}(S,E)$ and $\tilde{G}_{f_{0}}^{n_{0}}(\check{S%
},\check{E})$ is greater or equal to 2.
\end{proof}

\subsection{Qualitative property of
chaotic iterations}\label{sec:chaos-security-qualitative}

\begin{definition}[Topological mixing]
A discrete dynamical system is said to be topologically mixing
if and only if, for any couple of disjoint open set $U, V \neq \varnothing$,
$n_0 \in \mathds{N}$ can be found so that $\forall n \geqslant n_0, f^n(U) \cap
V \neq \varnothing$.
\end{definition}
\begin{proposition}
$\tilde{G}_{f_0}$ is topologically mixing on $(\mathcal{X}', d')$.
\end{proposition}

This result is an immediate consequence of the lemma below.

\begin{lemma}
For any open ball $B$ of $\mathcal{X}'$, an index $n$ can be found such that $\tilde{G}_{f_0}^n(B) = \mathcal{X}'$.
\end{lemma}

\begin{proof}
Let $B=B((E,S),\varepsilon)$ be an open ball, which the radius can be considered as strictly less than 1.
All the elements of $B$ have the same state $E$ and are such that an integer $k \left(=-\log_{10}(\varepsilon)\right)$ satisfies:
\begin{itemize}
\item all the strategies of $B$ have the same $k$ first terms,
\item after the index $k$, all values are possible.
\end{itemize}

Then, after $k$ iterations, the new state of the system is $\tilde{G}_{f_0}^k(E,S)_1$ and all the strategies are possible (all the points $(\tilde{G}_{f_0}^k(E,S)_1,\textrm{\^{S}})$, with any $\textrm{\^{S}} \in \mathbb{S}$, are reachable from $B$).

We will prove that all points of $\mathcal{X}'$ are reachable from $B$. Let
$(E',S') \in \mathcal{X}'$ and $s_i$ be the list of the different cells between
$\tilde{G}_{f_0}^k(E,S)_1$ and $E'$. We denote by $|s|$ the size of the sequence
$s_i$. So the point $(\check{E},\check{S})$ of $B$ defined by: $\check{E} = E$,
$\check{S}^i = S^i, \forall i \leqslant k$, $\check{S}^{k+i} = s_i, \forall i
\leqslant |s|$, and $\forall i \in \mathds{N}, S^{k + |s| + i} = S'^i$
is such that $\tilde{G}_{f_0}^{k+|s|}(\check{E},\check{S}) = (E',S')$. This concludes the proofs of the lemma and of the proposition.
\end{proof}



\subsection{Comparison between spread-spectrum and chaotic iterations}\label{sec:comparison-application-context}

The consequences of topological mixing for data hiding are multiple. Firstly, 
security can be largely improved by considering
the number of iterations as a secret key. An attacker
will reach all of the possible media when iterating without this key.
Additionally, he cannot benefit from a KOA setup, by studying media in the
neighborhood of the original cover. Moreover, as in a topological mixing
situation, it is possible that any hidden message (the initial condition), is
sent to the same fixed watermarked content (with different numbers of
iterations), the interest to be in a KMA setup is drastically reduced. Lastly, as
all of the watermarked contents are possible for a given hidden message, depending
on the number of iterations, CMA attacks will fail.


%Already corrected
The property of expansivity reinforces drastically the sensitivity in the aims of
reducing the benefits that Eve can obtain from an attack in KMA or KOA setup. For
example, it is impossible to have an estimation of the watermark by moving the
message (or the cover) as a cursor in situation of expansivity: this cursor will
be too much sensitive and the changes will be too important to be useful. On the
contrary, a very large constant of expansivity $\varepsilon$ is unsuitable: the
cover media will be strongly altered whereas the watermark would be undetectable.


% %Indeed, let us consider the same cover $E$ twice with two different watermarks
% $S,S'$. Thus $d(X,Y) < 1$, where $X=(E,S), Y=(E,S')$ and $d$ is for the distance
% previously defined. However, due to expansivity, $\exists n \in \mathds{N},
% d\left(G^n (X) ; G^n (Y) \right) \geqslant \varepsilon.$ Thus, $d_\infty
% \left(G^n (X)_1 ; G^n (Y)_1 \right) \geqslant \varepsilon - 1$, so either
% $d_\infty \left(X_1 ; G^n (X)_1 \right) \geqslant \dfrac{\varepsilon - 1}{2}$, or
% $d_\infty \left(Y_1 ; G^n (Y)_1 \right) \geqslant \dfrac{\varepsilon - 1}{2}$. If
% $\varepsilon$ is large, then at least one of the two watermarked media will be
% very different than its original cover.

Finally, spread-spectrum is relevant when
a discrete and secure data hiding technique is required in WOA setup. However,
this technique should not be used in KOA and KMA setup, due to its lack of
expansivity.% In the next section, we will present a new class of data hiding
schemes, which are expansive.


\section{Study of robustness: $CI_1$}\label{section:robustness-ci-1}

\subsection{Presentation of the architecture}\label{section:architecture-presentation}

In what follows, computations have been
performed on the supercomputer facilities of the M\'{e}socentre de calcul de Franche-Comt\'{e}.

In order to take benefits of parallelism, we have used Jace \cite{bhm09:ip}, a grid-enabled programming and execution
environment allowing a simple and efficient implementation 
of parallel and distributed applications. Roughly speaking, Jace builds a virtual parallel machine by 
connecting a set of heterogeneous and distant computers.
It schedules tasks, executes them, and returns results to the user. It
also proposes a simple programming interface for the implementation of applications
using a message passing model.

In our case, tasks are independent Python programs with different parameters. Jace takes care to execute them in parallel ensuring optimal load balancing and fault tolerance.

For these experiments we have used HPC resources 
from M\'{e}socentre of Franche-Comt\'{e}. This platform 
is currently composed of 720 cores with 11 TeraFLOPS of power.
The different tests have been realized on several images taken from the Wikimedia Commons repository~\cite{wiki:wikimedia-commons}.
The configurations of all tests are embedded into each graph caption.

%je ne sais pas comment expliquer vos configs !
%In the following experiments, we have chosen different configuration files,
%about 1000 images from wikimedia.....



\subsection{Concerning Tests}\label{sec:tests-realized-ci-1}

In this section, we intend to study the robustness of the $CI_1$ watermarking process. 
To achieve this goal, two batteries of tests have been executed.

In the first battery, we have used the same embedding key, thus a constant strategy has been used. 
In the second battery, random strategies have been used, in order to show the influence of the embedding key on robustness. 
The objective of these tests is to evaluate the correlation linking the parameters of some given geometrical attacks on the one hand, and the bits difference rate between the original watermark and its extracted version after attacks on the other hand. 
A fixed strategy is then used in these tests.

The following results have been obtained by using a large number of attacks on 600 images. 
For each category of attacks, two graphs have been plotted: the first one is about averages, when the second one is for the standard deviation.
These graphs help to determine the threshold leading to the acceptation/rejection of a possibly watermarked media.


\subsection{Tests results}\label{sec:tests-results-ci-1}

The exhaustive list of the geometrical attacks tested and the
corresponding graphs results is following:

\begin{enumerate}
  \item \textbf{Robustness facing a resizing attack.}\\
  See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
  \item \textbf{Robustness facing a JPEG compression
  attack.}\\See
  Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
  \item \textbf{Robustness facing a Gaussian blur
  attack.}\\See
  Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
  \item \textbf{Robustness facing a rotation
  attack.}\\See
  Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
  \item \textbf{Robustness facing a  blur
  attack.}\\See
  Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
  \item \textbf{Robustness facing a contrast
  attack.}\\See
  Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
  \item \textbf{Robustness facing a cropping
  attack.}\\See
  Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
\end{enumerate}


% \begin{figure*}
% \begin{center}
% %\includegraphics[width=10.5cm]{./img/prisoner-problem.png}
% \includegraphics[width=15cm]{./img/prisoner-problem}
% \end{center}
% 
% \caption{Simmons' prisoner problem~\cite{Simmons83}}
% \end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a resizing attack.(600 images and constant strategy)}
\label{fig:resizing-attack-constant-strategy}
\end{figure*}

\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a JPEG compression attack.(600 images and
constant strategy)}
\label{fig:jpeg-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a Gaussian blur attack.(600 images and
constant strategy)}
\label{fig:gaussian-blur-attack-constant-strategy}
\end{figure*}

\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a rotation attack.(600 images and constant
strategy)}
\label{fig:rotation-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a blur attack.(600 images and constant
strategy)}
\label{fig:blur-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a contrast attack.(600 images and constant
strategy)}
\label{fig:contrast-attack-constant-strategy}
\end{figure*}


\begin{figure*}[htb]
\begin{minipage}[b]{.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-average}}
    \centerline{(a) Average}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
  \centering
    \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-sd}}
    \centerline{(b) Standard deviation}
\end{minipage}
\caption{Robustness of $CI_1$ facing a cropping attack.(600 images and constant
strategy)}
\label{fig:cropping-attack-constant-strategy}
\end{figure*}

\subsection{Robustness evaluation}\label{sec:tests-results-ci-1}

From the results presented in the previous section, the robustness of the
process $CI_1$ can be evaluated.

\textbf{TODO: The interpretation which can be given of the robustness tests is
 that\ldots}



To conclude, at this point, only two information hiding schemes are both stego-secure and chaos-secure \cite{gfb10:ip}.
The first one is based on a spread spectrum technique called Natural Watermarking.
It is stego-secure when its parameter $\eta$ is equal to $1$ \cite{Cayre2008}. 
Unfortunately, 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 expansivity \cite{gfb10:ip}.
The second scheme both chaos-secure and stego-secure is based on chaotic iterations \cite{guyeux10ter}.
However, its first versions called $CI_1$ allows to embed securely only one bit per embedding parameters. 
The objective of the next sections is to improve the $CI_1$ scheme presented in \cite{guyeux10ter}, in such a way that more than one bit can be embedded.


\section{Improved process for steganography: $CI_2$}
\label{section:process-ci2}

\noindent In this section is given a new algorithm that generalize the
$CI_1$ scheme presented above. 

Let us firstly introduce the following notations:

\subsection{Notations}
\begin{itemize}
  %\item Let $\mathbb{S}_k=\llbracket 0, \mathsf{k-1} \rrbracket^{\mathds{N}}$ be
  %the set of all strategies with values in $\llbracket 0, \mathsf{k-1}
  %\rrbracket$.
  \item $x^0 \in \mathbb{B}^\mathsf{N}$ is the $\mathsf{N}$ least
  significant coefficients of a given cover media $C$.% $X$ be the initial state $X_0$,
  %\item The vectorial negation $f_0$ (defined in~\ref{Def:vectorial-negation})
  %as iteration function.
  \item $m^0 \in \mathbb{B}^\mathsf{P}$ is the watermark to embed into $x^0$.
  \item $S_1 \in \mathbb{S}_N$ is a strategy called \textbf{place strategy}.
  \item $S_2 \in \mathbb{S}_P$ is a strategy called \textbf{choice strategy}.
  \item Lastly, $S_3 \in \mathbb{S}_P$ is a strategy called \textbf{mixing strategy}.
\end{itemize}

%\subsection{Algorithm SCISMM in details}\label{sec:algo-SCISMM}

\subsection{Presentation of the scheme}

Our information hiding scheme called Steganography by Chaotic Iterations and
Substitution with Mixing Message (SCISMM) is defined by
%\begin{definition}[SCISMM]\label{def:algo}
$\forall (n,i,j) \in
\mathds{N}^{\ast} \times \llbracket 0;\mathsf{N-1}\rrbracket \times \llbracket
0;\mathsf{P-1}\rrbracket$:

\begin{equation*}
\left\{
\begin{array}{l}
x_i^n=\left\{
\begin{array}{ll}
x_i^{n-1} & \text{ if }S_1^n\neq i \\
m_{S_2^n} & \text{ if }S_1^n=i.
\end{array}
\right.
\\
\\
m_j^n=\left\{
\begin{array}{ll}
m_j^{n-1} & \text{ if }S_3^n\neq j \\
 & \\
\overline{m_j^{n-1}} & \text{ if }S_3^n=j.
\end{array}
\right.
\end{array}
\right.
\end{equation*}
%\end{definition}
where $\overline{m_j^{n-1}}$ is the boolean negation of $m_j^{n-1}$.
%\subsection{Stego-content by SCISMM}\label{sec:stego-content}

The stego-content is the boolean vector $y = x^P \in
\mathbb{B}^\mathsf{N}$.


\section{Study of security of $CI_2$}

\subsection{Study of stego-security}\label{sec:stego-security-ci-2}


\noindent Let us prove that,

\begin{proposition}
SCISMM is stego-secure.
\end{proposition}

\begin{proof}
%We suppose that $K,M \sim
%\mathbf{U}\left([0;1]\right)$, and $X \sim
%We consider a strategy in the CIIS scheme, and we
Let us suppose that $x^0 \sim
\mathbf{U}\left(\mathbb{B}^N\right)$ and $m^0 \sim
\mathbf{U}\left(\mathbb{B}^P\right)$ in a SCISMM setup. 
We will prove by a mathematical induction that $\forall n \in \mathds{N}, x^n
\sim \mathbf{U}\left(\mathbb{B}^N\right)$. The base case is obvious according
to the uniform repartition hypothesis. 

Let us now suppose that the statement $x^n \sim
\mathbf{U}\left(\mathbb{B}^N\right)$ holds for some $n$. 
For a given $k  \in \mathbb{B}^N$,
we denote by $\tilde{k_i} \in \mathbb{B}^N$ the vector defined by: $\forall i
\in  \llbracket 0;\mathsf{N-1}\rrbracket,$
if
$k=\left(k_0,k_1,\hdots,k_i,\hdots,k_{\mathsf{N}-2},k_{\mathsf{N}-1}\right)$,\newline
then $\tilde{k}_i=\left( k_0,k_1,\hdots,\overline{k_i},\hdots,k_{\mathsf{N}-2},k_{\mathsf{N}-1} \right).$


%\begin{equation*}
Let
$E_{i,j}$ be the following events: $$\forall (i,j) \in \llbracket
0;\mathsf{N-1}\rrbracket \times \llbracket 0;\mathsf{P-1}\rrbracket, E_{i,j}=$$
$$S_1^{n+1}=i \wedge S_2^{n+1}=j \wedge  m_j^{n+1}=k_i \wedge \left(x^n=k \vee x^n=\tilde{k}_i\right),$$
and
$p=P\left(x^{n+1}=k\right).$
So,
\begin{equation*}
p=P\left(\bigvee_{i \in \llbracket
0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
E_{i,j} \right).
\end{equation*}
We now introduce the following notation: $P_1(i)=P\left(S_1^{n+1}=i\right),$
 $P_2(j)=P\left(S_2^{n+1}=j\right),$
 $P_3(i,j)=P\left(m_j^{n+1}=k_i\right),$
and $P_4(i)=P\left(x^n=k \vee x^n=\tilde{k}_i\right).$


These four events are independent in SCISMM setup, thus:
\begin{equation*}
p=\sum_{i \in \llbracket
0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
P_1(i)P_2(i)P_3(i,j)P_4(i).
\end{equation*}
According to 
Proposition~\ref{prop:CIIS-stego-security}, $P\left(m_j^{n+1}=k_i\right)=\frac{1}{2}$.
As the two events are incompatible:
$$P\left(x^n=k \vee x^n=\tilde{k}_i\right)=P\left(x^n=k\right) + P\left(x^n=\tilde{k}_i\right).$$


Then, by using the inductive hypothesis:
$P\left(x^n=k\right)=\frac{1}{2^N},$
and
$P\left(x^n=\tilde{k}_i\right)=\frac{1}{2^N}.$

Let $S$ be defined by $$S=\sum_{i
\in \llbracket 0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
P_1(i)P_2(j).$$
Then $p =  2 \times \frac{1}{2} \times \frac{1}{2^N} \times S = \frac{1}{2^N} \times S$.

$S$ can now be evaluated:
\begin{equation*}
\begin{array}{lll}
S & = &  \sum_{i
\in \llbracket 0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
P_1(i)P_2(j)\\
  & = &  \sum_{i \in \llbracket 0;\mathsf{N-1}\rrbracket} P_1(i) \times \sum_{j \in \llbracket 0;\mathsf{P-1}\rrbracket}
P_2(j).\\
\end{array}
\end{equation*}

The set of events $\left \{ S_1^{n+1}=i \right \}$ for $i \in \llbracket 0;N-1
\rrbracket$ and the set of events $\left \{ S_2^{n+1}=j \right \}$ for $j \in
\llbracket 0;P-1 \rrbracket$ are both a partition of the universe of possible,
so $S=1$.

%Evaluate now $ P\left(S^n=k\right)$.
%\newline
%We have: $K^0 = M \otimes K \sim
%\mathbf{U}\left([0;1]\right)$, so $K^\prime=F\left( K^0,p \right) \sim
%\mathbf{U}\left([0;1]\right)$, because for PLCMs, ``Uniform input generate
%uniform output''~\cite{Lasota}.
%Well with an immediate proof by recurrence we have:
%$K^n \sim \mathbf{U}\left([0;1]\right)$. As $S^n = \left \lfloor N \times X^n
%\right \rfloor + 1$ we can deduce that: $S^n \sim
%\mathbf{U}\left([1;N]\right)$. Therefore: $ P\left(S^n=k\right)=\frac{1}{N}$.
Finally, 
%\begin{equation*}
%\begin{array}{llr}
%P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
% P\left(S^n=k\right) & \\
% & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} & \\
% & =\frac{1}{2^N} & \\
%\end{array}
%\end{equation*}
%\begin{equation*}
%$\begin{array}{llr}
%P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
% P\left(S^n=k\right) & \\
%P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} &
% =\frac{1}{2^N} \\
%\end{array}$.
$P\left(x^{n+1}=k\right)=\frac{1}{2^N}$, which leads to $x^{n+1} \sim
\mathbf{U}\left(\mathbb{B}^N\right)$.
This result is true $\forall n \in \mathds{N}$, we thus have proven that the
stego-content $y$ is uniform in the set of possible stego-content, so
$y \sim \mathbf{U}\left(\mathbb{B}^N\right) \text{
when } x \sim \mathbf{U}\left(\mathbb{B}^N\right).$
\end{proof}


\subsection{Study of chaos security of $CI_2$}\label{sec:chaos-security-ci2}
\subsubsection{Topological model}
\label{sec:SCISMM-topological-model}

%\section{THE NEW TOPOLOGICAL SPACE}
\noindent In this section, we prove that SCISMM can be modeled 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.


\paragraph{Iteration Function and Phase Space}\label{Defining}


%\begin{definition}[Function $F$]
Let
\begin{equation*}
\begin{array}{ll}
F: & \llbracket0;\mathsf{N-1}\rrbracket \times
\mathds{B}^{\mathsf{N}}\times \llbracket0;\mathsf{P-1}\rrbracket  \times
\mathds{B}^{\mathsf{P}}
\longrightarrow \mathds{B}^{\mathsf{N}} \\
 & (k,x,\lambda,m)  \longmapsto  \left( \delta
 (k,j).x_j+\overline{\delta (k,j)}.m_{\lambda}\right) _{j\in
 \llbracket0;\mathsf{N-1}\rrbracket}\\
 \end{array}%
\end{equation*}%
%\end{definition}

\noindent where + and . are the boolean addition and product operations.

Consider the phase space $\mathcal{X}_2$ defined as follow:

%\begin{definition}[Phase space $\mathcal{X}_2$]

\begin{equation*}
\mathcal{X}_2 =  \mathbb{S}_N \times \mathds{B}^\mathsf{N} \times \mathbb{S}_P
\times \mathds{B}^\mathsf{P} \times \mathbb{S}_P,
\end{equation*}
where $\mathbb{S}_N$ and $\mathbb{S}_P$ are the sets introduced in Section \ref{section:Algorithm}.
%\end{definition}

We define the map $\mathcal{G}_{f_0}:\mathcal{X}_2  \longrightarrow  \mathcal{X}_2$ by:

\begin{equation*}
\mathcal{G}_{f_0}\left(S_1,x,S_2,m,S_3\right) =   
\end{equation*}
\begin{equation*}
\left(\sigma_N(S_1),
F(i_N(S_1),x,i_P(S_2),m),\sigma_P(S_2),G_{f_0}(m,S_3),\sigma_P(S_3)\right)
\end{equation*}

\noindent Then SCISMM can be described by the
iterations of the following discret dynamical system:

%\begin{definition}[Discret Dynamical System for SCISMM]
\begin{equation*}
\left\{
\begin{array}{l}
X^0 \in \mathcal{X}_2 \\
X^{k+1}=\mathcal{G}_{f_0}(X^k).%
\end{array}%
\right.
\end{equation*}%
%\end{definition}%


\paragraph{Cardinality of $\mathcal{X}_2$}

By comparing $\mathcal{X}_2$ and $\mathcal{X}_1$, we have the following result.

\begin{proposition}
The phase space $\mathcal{X}_2$ has, at least, the cardinality of the continuum.
\end{proposition}

\begin{proof}
Let $\varphi$ be the map defined as follow:

\begin{equation*}
\begin{array}{lcll}
\varphi: & \mathcal{X}_1 & \longrightarrow & \mathcal{X}_2 \\
 & (S,x) &  \longmapsto  &(S,x,0,0,0)\\
 \end{array}%
\end{equation*}%

\noindent $\varphi$ is  injective.
So the cardinality of  $\mathcal{X}_2$ is greater than or equal to the
cardinality of $\mathcal{X}_1$. And consequently
$\mathcal{X}_2$ has at least the cardinality of the continuum.
\end{proof}


\begin{remark}
This result is independent on the number of cells of the system.
\end{remark}


\paragraph{A New Distance on $\mathcal{X}_2$}

We define a new distance on $\mathcal{X}_2$ as follow: 
%\begin{definition}[Distance $d_2$ on $\mathcal{X}_2$]
$\forall X,\check{X} \in \mathcal{X}_2,$ if $X =(S_1,x,S_2,m,S_3)$ and $\check{X} =
(\check{S_1},\check{x},\check{S_2},\check{m}, \check{S_3}),$ then:
\begin{equation*}
\begin{array}{lll}
d_2(X,\check{X}) & = &
d_{\mathds{B}^{\mathsf{N}}}(x,\check{x})+d_{\mathds{B}^{\mathsf{P}}}(m,\check{m})\\
& + &
d_{\mathbb{S}_N}(S_1,\check{S_1})+d_{\mathbb{S}_P}(S_2,\check{S_2})+d_{\mathbb{S}_P}(S_3,\check{S_3}),\\
\end{array}
\end{equation*}
where $d_{\mathds{B}^{\mathsf{N}}}$, $d_{\mathds{B}^{\mathsf{P}}}$,
$d_{\mathbb{S}_N}$, and $d_{\mathbb{S}_P}$ are the same distances than in
Definition~\ref{def:distance-on-X1}.
%\end{definition}






\subsubsection{Continuity of $CI_2$}

To prove that SCISMM is another example of topological chaos in the
sense of Devaney, $\mathcal{G}_{f_0}$ must be continuous on the metric
space $(\mathcal{X}_2,d_2)$.

\begin{proposition}
$\mathcal{G}_{f_0}$ is a continuous function on $(\mathcal{X}_2,d_2)$.
\end{proposition}

\begin{proof}
We use the sequential continuity.

Let $((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n)_{n\in \mathds{N}}$ be a sequence of the
phase space $\mathcal{X}_2$, which converges to $(S_1,x,S_2,m,S_3)$. We will prove
that $\left( \mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n)\right) _{n\in
\mathds{N}}$ converges to $\mathcal{G}_{f_0}(S_1,x,S_2,m,S_3)$. 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 (\emph{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
$d_{\mathds{B}^{\mathsf{N}}}(x^n,x)$, $d_{\mathds{B}^{\mathsf{P}}}(m^n,m)$,  
$d_{\mathbb{S}_N}((S_1)^n,S_1)$, $d_{\mathbb{S}_P}((S_2)^n,S_2)$, and
$d_{\mathbb{S}_P}((S_3)^n,S_3)$ converges to 0. But
$d_{\mathds{B}^{\mathsf{N}}}(x^n,x)$ and $d_{\mathds{B}^{\mathsf{P}}}(m^n,m)$
are integers, so $\exists n_{0}\in \mathds{N}, \forall
n \geqslant n_0,
d_{\mathds{B}^{\mathsf{N}}}(x^n,x)=0$ and $\exists n_{1}\in \mathds{N},
\forall n \geqslant n_1, d_{\mathds{B}^{\mathsf{P}}}(m^n,m)=0$. 

Let $n_3=Max(n_0,n_1)$.
In other words, there exists a threshold $n_{3}\in \mathds{N}$ after which no
cell will change its state:
$\exists n_{3}\in \mathds{N},n\geqslant n_{3}\Longrightarrow (x^n = x) \wedge
(m^n = m).$

In addition, $d_{\mathbb{S}_N}((S_1)^n,S_1)\longrightarrow 0$,
$d_{\mathbb{S}_P}((S_2)^n,S_2)\longrightarrow 0$, and
$d_{\mathbb{S}_P}((S_3)^n,S_3)\longrightarrow 0$, so $\exists n_4,n_5,n_6\in
 \mathds{N},$
 \begin{itemize}
   \item $\forall n \geqslant n_4, d_{\mathbb{S}_N}((S_1)^n,S_1)<10^{-1}$,
   \item $\forall n \geqslant n_5, d_{\mathbb{S}_P}((S_2)^n,S_2)<10^{-1}$,
   \item $\forall n \geqslant n_6, d_{\mathbb{S}_P}((S_3)^n,S_3)<10^{-1}$.
 \end{itemize}
 
 Let $n_7=Max(n_4,n_5,n_6)$. For
 $n\geqslant n_7$, all the strategies $(S_1)^n$, $(S_2)^n$, and $(S_3)^n$ have the same
 first term, which are respectively $(S_1)_0$,$(S_2)_0$ and $(S_3)_0$ :$\forall n\geqslant n_7,$
$$((S_1)_0^n={(S_1)}_0)\wedge
((S_2)_0^n={(S_2)}_0)\wedge ((S_3)_0^n={(S_3)}_0).
$$

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
$\mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n)$ and $\mathcal{G}_{f_0}(S_1,x,S_2,m,S_3)$ 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 between $\left(
\mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n)\right) $ and $\left(
\mathcal{G}_{f_0}(S_1,x,S_2,m,S_3)\right) $ is convergent to 0.
Let $\varepsilon >0$. 

\begin{itemize}
\item If $\varepsilon \geqslant 1$, we have seen that distance
between $\mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n)$ and 
$\mathcal{G}_{f_0}(S_1,x,S_2,m,S_3)$ is
strictly less than 1 after the $(n_8)^{th}$ term (same state).
\medskip

\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
\frac{\varepsilon}{3} \geqslant 10^{-(k+1)}$. As
$d_{\mathbb{S}_N}((S_1)^n,S_1)$, $d_{\mathbb{S}_P}((S_2)^n,S_2)$ and
$d_{\mathbb{S}_P}((S_3)^n,S_3)$  converges to 0, we have:
\begin{itemize}
 \item $\exists n_9\in \mathds{N},\forall n\geqslant
 n_9,d_{\mathbb{S}_N}((S_1)^n,S_1)<10^{-(k+2)}$,
 \item $\exists n_{10}\in \mathds{N},\forall n\geqslant
 n_{10},d_{\mathbb{S}_P}((S_2)^n,S_2)<10^{-(k+2)}$,
 \item $\exists n_{11}\in \mathds{N},\forall n\geqslant
 n_{11},d_{\mathbb{S}_P}((S_3)^n,S_3)<10^{-(k+2)}$.
\end{itemize}
  
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.
\end{itemize}

\noindent As a consequence, the $k+1$ first entries of the strategies of
$\mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n)$ and $
\mathcal{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
$d_{\mathbb{S}_N}$ and $d_{\mathbb{S}_P}$:
$$d_2\left(\mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n);\mathcal{G}_{f_0}(S_1,x,S_2,m,S_3)\right)$$
is equal to :
$$d_{\mathbb{S}_N}((S_1)^n,S_1) + d_{\mathbb{S}_P}((S_2)^n,S_2) +
d_{\mathbb{S}_P}((S_3)^n,S_3)$$
which is smaller than $3.10^{-(k+1)} \leqslant 3.\frac{\varepsilon}{3} =\varepsilon$.

Let $N_{0}=max(n_{8},n_{12})$. We can claim that
\begin{equation*}
\forall \varepsilon >0,\exists N_{0}\in \mathds{N},\forall n\geqslant N_{0},
\end{equation*}
\begin{equation*}
d_2\left(\mathcal{G}_{f_0}((S_1)^n,x^n,(S_2)^n,m^n,(S_3)^n);\mathcal{G}_{f_0}(S_1,x,S_2,m,S_3)\right)
\leqslant \varepsilon .
\end{equation*}
$\mathcal{G}_{f_0}$ is consequently continuous on $(\mathcal{X}_2,d_2)$.
\end{proof}



% >>>>>>>>>>>>>>>>>>>>>> Discrete chaotic iterations as topological chaos <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
%\subsection{Study of chaos-security}\label{section:chaos-security-ci-2}

\noindent To prove that we are in the framework of Devaney's topological chaos,
we have to check the regularity, transitivity, and sensitivity conditions.
%It will be proven that $\mathcal{G}_{f_0}$ function as introduce in
%definition~\ref{} satisfies these three hypotheses.



\subsubsection{Regularity}

\begin{proposition}%[Regularity of $\mathcal{G}_{f_0}$]
\label{prop:regularite}
Periodic points of $\mathcal{G}_{f_0}$ are dense in $\mathcal{X}_2$.
\end{proposition}

\begin{proof}
Let $(\check{S_1},\check{x},\check{S_2},\check{m}, \check{S_3})\in
\mathcal{X}_2$ and $\varepsilon >0$. We are looking for a periodic point
$(\widetilde{S_1},\widetilde{x},\widetilde{S_2},\widetilde{m}, \widetilde{S_3})$
satisfying $d_2((\check{S_1},\check{x},\check{S_2},\check{m},
\check{S_3});(\widetilde{S_1},\widetilde{x},\widetilde{S_2},\widetilde{m}, \widetilde{S_3}))<\varepsilon$.

As $\varepsilon$ can be strictly lesser than 1, we must choose
$\widetilde{x} = \check{x}$ and
$\widetilde{m} = \check{m}$.
Let us define $k_0(\varepsilon) =\lfloor
- log_{10}(\frac{\varepsilon}{3} )\rfloor +1$ and consider the set:
$\mathcal{S}_{\check{S_1},\check{S_2},\check{S_3}, k_0(\varepsilon)} =$
$\left\{ S
\in \mathbb{S}_N \times \mathbb{S}_P \times \mathbb{S}_P/ ((S_1)^k =
\check{S_1}^k) \wedge ((S_2)^k =\check{S_2}^k))\right.$
$\left.\wedge ((S_3)^k=\check{S_3}^k)), \forall k \leqslant k_0(\varepsilon)
\right\}$.


Then, 
$\forall (S_1,S_2,S_3) \in
\mathcal{S}_{\check{S_1},\check{S_2},\check{S_3},k_0(\varepsilon)},$
$d_2((S_1,\check{x},S_2,\check{m},S_3);(\check{S_1},\check{x},\check{S_2},\check{m},
\check{S_3})) < 3.\frac{\varepsilon}{3}=\varepsilon.$
It remains to choose $(\widetilde{S_1},\widetilde{S_1},\widetilde{S_1}) \in
\mathcal{S}_{\check{S_1},\check{S_2},\check{S_3}, k_0(\varepsilon)}$ such that
$(\widetilde{S_1},\widetilde{x},\widetilde{S_2},\widetilde{m}, \widetilde{S_3})
= (\widetilde{S_1},\check{x},\widetilde{S_2},\check{m}, \widetilde{S_3})$ is
a periodic point for $\mathcal{G}_{f_0}$.

Let $\mathcal{J} =$
$\left\{ i \in \llbracket0;\mathsf{N-1}\rrbracket / x_i \neq
\check{x_i}, \text{ where }\right.$
$\left.(S_1,x,S_2,m,S_3) = \mathcal{G}_{f_0}^{k_0}
(\check{S_1},\check{x},\check{S_2},\check{m},\check{S_3}) \right\},$
$\lambda = card(\mathcal{J})$, and
$j_0 <j_1 < ... < j_{\lambda-1}$ the elements of $ \mathcal{J}$. 

\begin{enumerate}
  \item Let us firstly build three strategies: $S_1^\ast$,  $S_2^\ast$, and 
  $S_3^\ast$, as follows.
\begin{enumerate}
  \item 
$(S_1^\ast)^k=\check{S_1}^k$, $(S_2^\ast)^k=\check{S_2}^k$, and
$(S_3^\ast)^k=\check{S_3}^k$, if $k \leqslant k_0(\varepsilon).$ 
\item Let us now explain how to replace $\check{x}_{j_q}$, $\forall q \in
  \llbracket0;\mathsf{\lambda-1}\rrbracket$:\newline
First of all, we must replace $\check{x}_{j_0}$:
\begin{enumerate}
  \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
  \check{x}_{j_0}=m_{\lambda_0}$, then we can choose
 $(S_1^\ast)^{k_0+1}=j_0$, $(S_2^\ast)^{k_0+1}=\lambda_0$,
 $(S_3^\ast)^{k_0+1}=\lambda_0$, and so $I_{j_0}$ will be equal to $1$.
 \item If such a $\lambda_0$ does not exist, we choose:\newline $(S_1^\ast)^{k_0+1}=j_0$, $(S_2^\ast)^{k_0+1}=0$,
 $(S_3^\ast)^{k_0+1}=0$,\newline $(S_1^\ast)^{k_0+2}=j_0$,
 $(S_2^\ast)^{k_0+2}=0$, $(S_3^\ast)^{k_0+2}=0$, \newline and 
 $I_{j_0}=2$.\newline

All of the $\check{x}_{j_q}$ are replaced similarly. 
The other terms of $S_1^\ast$,  $S_2^\ast$, and 
  $S_3^\ast$ are constructed identically, and the values of $I_{j_q}$ are defined in
  the same way.\newline
 Let $\gamma = \sum_{q=0}^{\lambda-1}I_{j_q}$.
\end{enumerate}

\item Finally, let $(S_1^\ast)^{k}=(S_1^\ast)^{j}$, $(S_2^\ast)^{k}=(S_2^\ast)^{j}$,
and $(S_3^\ast)^{k}=(S_3^\ast)^{j}$, where $j\leqslant k_{0}(\varepsilon )+\gamma$
is satisfying $j\equiv k~[\text{mod } (k_{0}(\varepsilon )+\gamma)]$, if
$k>k_{0}(\varepsilon )+\gamma$.
\end{enumerate}
So,
$\mathcal{G}_{f_0}^{k_{0}(\varepsilon
)+\gamma}(S_1^\ast,\check{x},S_2^\ast,\check{m},S_3^\ast)=(S_1^\ast,\check{x},S_2^\ast,m,S_3^\ast).$
Let $\mathcal{K} =\left\{ i \in \llbracket0;\mathsf{P-1}\rrbracket / m_i \neq
\check{m_i}, \text{ where }\right.$\newline
$\left.\mathcal{G}_{f_0}^{k_{0}(\varepsilon
)+\gamma}(S_1^\ast,\check{x},S_2^\ast,\check{m},S_3^\ast)=(S_1^\ast,\check{x},S_2^\ast,m,S_3^\ast)
\right\},$\newline
$\mu = card(\mathcal{K})$, and
$r_0 <r_1 < ... < r_{\mu-1}$ the elements of $ \mathcal{K}$.  
 
 
 
  \item Let us now build the strategies $\widetilde{S_1}$, 
  $\widetilde{S_2}$, $\widetilde{S_3}$.
\begin{enumerate}
  \item Firstly, let $\widetilde{S_1}^k=(S_1^\ast)^k$, $\widetilde{S_2}^k=(S_2^\ast)^k$, and
$\widetilde{S_3}^k=(S_3^\ast)^k$, if $k \leqslant k_0(\varepsilon) + \gamma.$
\item  How to replace $\check{m}_{r_q}, \forall q \in
  \llbracket0;\mathsf{\mu-1}\rrbracket$:\newline
  
First of all, let us explain how to replace $\check{m}_{r_0}$:
\begin{enumerate}
  \item If $\exists \mu_0 \in \llbracket0;\mathsf{N-1}\rrbracket /
  \check{x}_{\mu_0}=m_{r_0}$, then 
 we can choose $\widetilde{S_1}^{k_0+\gamma +1}=\mu_0$, $\widetilde{S_2}^{k_0+\gamma
 +1}=r_0$, $\widetilde{S_2}^{k_0+\gamma
 +1}=r_0$.\newline In that situation, we define $J_{r_0}=1$.
 \item If such a $\mu_0$ does not exist, then we can choose:\newline
$\widetilde{S_1}^{k_0+\gamma +1}=0$, $\widetilde{S_2}^{k_0+\gamma
 +1}=r_0$, $\widetilde{S_2}^{k_0+\gamma
 +1}=r_0$,\newline 
$\widetilde{S_1}^{k_0+\gamma +2}=0$, $\widetilde{S_2}^{k_0+\gamma
 +2}=r_0$, $\widetilde{S_2}^{k_0+\gamma
 +2}=0$,\newline 
$\widetilde{S_1}^{k_0+\gamma +3}=0$, $\widetilde{S_2}^{k_0+\gamma
 +3}=r_0$, $\widetilde{S_2}^{k_0+\gamma
 +3}=0$.\newline 
 Let $J_{r_0}=3$.

Then the other $\check{m}_{r_q}$ are replaced as previously,
  the other terms of $\widetilde{S_1}$, 
  $\widetilde{S_2}$, and $\widetilde{S_3}$ are constructed in the same way, and the
  values of $J_{r_q}$ are defined similarly.\newline
 Let $\alpha = \sum_{q=0}^{\mu-1}J_{r_q}$.
\end{enumerate}
\item Finally, let $\widetilde{S_1}^{k}=\widetilde{S_1}^{j}$,
$\widetilde{S_2}^{k}=\widetilde{S_2}^{j}$, and
$\widetilde{S_3}^{k}=\widetilde{S_3}^{j}$ where $j\leqslant k_{0}(\varepsilon
)+\gamma +\alpha$ is satisfying $j\equiv k~[\text{mod } (k_{0}(\varepsilon
)+\gamma + \alpha)]$, if $k>k_{0}(\varepsilon )+\gamma + \alpha$.
\end{enumerate}
\end{enumerate}


So, $\mathcal{G}_{f_0}^{k_{0}(\varepsilon
)+\gamma+\alpha}(\widetilde{S_1},\check{x},\widetilde{S_2},\check{m},\widetilde{S_3})=(\widetilde{S_1},\check{x},\widetilde{S_2},\check{m},\widetilde{S_3})$
%\end{enumerate}



Then, $(\widetilde{S_1},\widetilde{S_2},\widetilde{S_3}) \in
\mathcal{S}_{\check{S_1},\check{S_2},\check{S_3},k_0(\varepsilon)}$ defined as
previous is such that $(\widetilde{S_3},\check{x},\widetilde{S_3},\check{m},\widetilde{S_3})$
 is a periodic point, of period $k_{0}(\varepsilon
)+\gamma+\alpha$, which is $\varepsilon -$close to $(\check{S_1},\check{x},\check{S_2},\check{m}, \check{S_3})$.

As a conclusion, $(\mathcal{X}_2,\mathcal{G}_{f_0})$ is regular.
\end{proof}

\subsubsection{Transitivity}\label{sec:transitivite}

\begin{proposition}%[Transitivity of $ \mathcal{G}_{f_0}$]
\label{prop:transitivite}
$(\mathcal{X}_2,\mathcal{G}_{f_0})$ is topologically transitive.
\end{proposition}

\begin{proof}
Let us define $\mathcal{X}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{N}},$
such that $\mathcal{X}(S_1,x,S_2,m,S_3)=x$ and
$\mathcal{M}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{P}},$ such that
$\mathcal{M}(S_1,x,S_2,m,S_3)=m$. Let \linebreak
$\mathcal{B}_{A}=\mathcal{B}(X_{A},r_{A}) $ and
$\mathcal{B}_{B}=\mathcal{B}(X_{B},r_{B})$ be two open balls of $\mathcal{X}_2$,
with\linebreak $X_{A}=((S_1)_{A},x_{A},(S_2)_{A},m_{A},(S_3)_{A})$ and
$X_{B}=((S_1)_{B},x_{B},(S_2)_{B},m_{B},(S_3)_{B})$. We are looking for
$\widetilde{X}=(\widetilde{S_1},\widetilde{x},\widetilde{S_2},\widetilde{m},\widetilde{S_3})$
in $\mathcal{B}_{A} $ such that $\exists n_{0}\in
\mathbb{N},\mathcal{G}_{f_0}^{n_{0}}(\widetilde{X})\in \mathcal{B}_{B}$.\newline
$\widetilde{X}$ must be in $\mathcal{B}_{A}$ and $r_{A}$ can be strictly
lesser than 1, so $\widetilde{x}=x_{A}$ and $\widetilde{m}=m_{A}$. Let
$k_{0}=\lfloor - \log _{10}(\frac{r_{A}}{3})+1\rfloor $.
Let us notice $\mathcal{S}_{X_{A}, k_0} =\left\{ (S_1,S_2,S_3)
\in \mathbb{S}_\mathsf{N}\times (\mathbb{S}_\mathsf{P})^2/ \forall k\leqslant k_{0},\right.$
$\left.(S_1^{k}=(S_1)_{A}^{k})\wedge
(S_2^{k}=(S_2)_{A}^{k})\wedge (S_3^{k}=(S_3)_{A}^{k}))\right\}$.

Then $\forall (S_1,S_2,S_3) \in \mathcal{S}_{X_{A}, k_0},
(S_1,\widetilde{x},S_2,\widetilde{m},S_3)\in \mathcal{B}_{A}.$


Let $\mathcal{J} =\left\{i\in
\llbracket0,\mathsf{N-1}\rrbracket/\check{x}_{i}\neq
\mathcal{X}(X_{B})_{i}, \text{ where }\right.$\newline
$\left.(\check{S_1},\check{x},\check{S_2},\check{m},\check{S_3})=\mathcal{G}_{f_0}^{k_{0}}(X_{A})\right\},$
$\lambda = card(\mathcal{J})$,\newline and
$j_0 <j_1 < ... < j_{\lambda-1}$ the elements of $ \mathcal{J}$.

\begin{enumerate}
  \item Let us firstly build three strategies: $S_1^\ast$,  $S_2^\ast$, and 
  $S_3^\ast$ as follows.
\begin{enumerate}
  \item 
$(S_1^\ast)^k=(S_1)_A^k$, $(S_2^\ast)^k=(S_2)_A^k$, and
$(S_3^\ast)^k=(S_3)_A^k$, if $k \leqslant k_0.$
\item Let us now explain how to replace $\mathcal{X}(X_{B})_{j_q}$, $\forall q \in
  \llbracket0;\mathsf{\lambda-1}\rrbracket$:\newline
First of all, we must replace $\mathcal{X}(X_{B})_{j_0}$:
\begin{enumerate}
  \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
  \mathcal{X}(X_{B})_{j_0}=\check{m}_{\lambda_0}$, then we can choose
 $(S_1^\ast)^{k_0+1}=j_0$, $(S_2^\ast)^{k_0+1}=\lambda_0$,
 $(S_3^\ast)^{k_0+1}=\lambda_0$, and so $I_{j_0}$ will be equal to $1$.
 \item If such a $\lambda_0$ does not exist, we choose:\newline
 $(S_1^\ast)^{k_0+1}=j_0$, $(S_2^\ast)^{k_0+1}=0$,
 $(S_3^\ast)^{k_0+1}=0$,\newline $(S_1^\ast)^{k_0+2}=j_0$,
 $(S_2^\ast)^{k_0+2}=0$, $(S_3^\ast)^{k_0+2}=0$ \newline and so let us notice
 $I_{j_0}=2$.\newline

All of the $\mathcal{X}(X_{B})_{j_q}$ are replaced similarly.
The other terms of $S_1^\ast$,  $S_2^\ast$, and $S_3^\ast$ are constructed identically, and the values of $I_{j_q}$ are defined on the same way.\newline
Let $\gamma = \sum_{q=0}^{\lambda-1}I_{j_q}$.
\end{enumerate}

\item $(S_1^\ast)^{k}=(S_1^\ast)^{j}$, $(S_2^\ast)^{k}=(S_2^\ast)^{j}$
and $(S_3^\ast)^{k}=(S_3^\ast)^{j}$ where $j\leqslant k_{0}+\gamma$
is satisfying $j\equiv k~[\text{mod } (k_{0}+\gamma)]$, if
$k>k_{0}+\gamma$.
\end{enumerate}
So,$\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_1^\ast,x_A,S_2^\ast,m_A,S_3^\ast))=(S_1^\ast,x_B,S_2^\ast,m,S_3^\ast)$

Let $\mathcal{K} =\left\{ i \in \llbracket0;\mathsf{P-1}\rrbracket / m_i \neq
\mathcal{M}(X_{B})_{i}, \text{ where }\right.$\newline
$\left.(S_1^\ast,x_B,S_2^\ast,m,S_3^\ast)=\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_1^\ast,x_A,S_2^\ast,m_A,S_3^\ast))\right\},$\newline
$\mu = card(\mathcal{K})$ and $r_0 <r_1 < ... < r_{\mu-1}$ the elements of $
\mathcal{K}$.
 
 
  \item Let us secondly build three other strategies:  $\widetilde{S_1}$, 
  $\widetilde{S_2}$, $\widetilde{S_3}$ as follows.
\begin{enumerate}
  \item $\widetilde{S_1}^k=(S_1^\ast)^k$, $\widetilde{S_2}^k=(S_2^\ast)^k$, and
$\widetilde{S_3}^k=(S_3^\ast)^k$, if $k \leqslant k_0 + \gamma.$
\item  Let us now explain how to replace $\mathcal{M}(X_{B})_{r_q}, \forall q
\in \llbracket0;\mathsf{\mu-1}\rrbracket$:
  
First of all, we must replace $\mathcal{M}(X_{B})_{r_0}$:
\begin{enumerate}
  \item If $\exists \mu_0 \in \llbracket0;\mathsf{N-1}\rrbracket /
  \mathcal{M}(X_{B})_{r_0}=(x_B)_{\mu_0}$, then we can choose 
 $\widetilde{S_1}^{k_0+\gamma +1}=\mu_0$, $\widetilde{S_2}^{k_0+\gamma
 +1}=r_0$, $\widetilde{S_3}^{k_0+\gamma
 +1}=r_0$, and $J_{r_0}$ will be equal to $1$.
 \item If such a $\mu_0$ does not exist, we choose:
$\widetilde{S_1}^{k_0+\gamma +1}=0$, $\widetilde{S_2}^{k_0+\gamma
 +1}=r_0$, $\widetilde{S_3}^{k_0+\gamma
 +1}=r_0$,\newline 
$\widetilde{S_1}^{k_0+\gamma +2}=0$, $\widetilde{S_2}^{k_0+\gamma
 +2}=r_0$, $\widetilde{S_3}^{k_0+\gamma
 +2}=0$,\newline 
$\widetilde{S_1}^{k_0+\gamma +3}=0$, $\widetilde{S_2}^{k_0+\gamma
 +3}=r_0$, $\widetilde{S_3}^{k_0+\gamma
 +3}=0$,\newline 
 and so let us notice $J_{r_0}=3$.\newline

All the $\mathcal{M}(X_{B})_{r_q}$ are replaced similarly.
The other terms of $\widetilde{S_1}$, 
$\widetilde{S_2}$, and $\widetilde{S_3}$ are constructed identically, and the
values of $J_{r_q}$ are defined on the same way.\newline
Let $\alpha = \sum_{q=0}^{\mu-1}J_{r_q}$.
\end{enumerate}
\item $\forall k \in \mathbb{N}^\ast, \widetilde{S_1}^{k_0+ \gamma + \alpha
+ k}=(S_1)_B^{k}$, $\widetilde{S_2}^{k_0+ \gamma + \alpha + k}=(S_2)_B^{k}$, and
$\widetilde{S_3}^{k_0+ \gamma + \alpha + k}=(S_3)_B^{k}$.
\end{enumerate}
\end{enumerate}

So,
$\mathcal{G}_{f_0}^{k_{0}+\gamma+\alpha}(\widetilde{S_1},x_A,\widetilde{S_2},m_A,\widetilde{S_3})=X_B$,
with $(\widetilde{S_1},\widetilde{S_2},\widetilde{S_3}) \in \mathcal{S}_{X_{A},
k_0}$.
Then $\widetilde{X} = (\widetilde{S_1},x_A,\widetilde{S_2},m_A,\widetilde{S_3})
\in \mathcal{X}_2$ is such that $\widetilde{X} \in \mathcal{B}_A$ and
$\mathcal{G}_{f_0}^{k_0+\gamma+\alpha}( \widetilde{X}) \in \mathcal{B}_B$.
Finally we have proven the result.
\end{proof}

\subsubsection{Sensibility to the Initial Condition}\label{sec:sensibilite}

\begin{proposition}%[Sensitivity of $,\mathcal{G}_{f_0}$]
\label{prop:sensibilite}
$(\mathcal{X}_2,\mathcal{G}_{f_0})$ has sensitive dependence on initial conditions.
\end{proposition}

\begin{proof}
$\mathcal{G}_{f_0}$ is regular and transitive. Due to Theorem~\ref{theo:banks}, $\mathcal{G}_{f_0}$ is sensitive.
\end{proof}

%In a following  section, we will show that this result can be stated independently of regularity,
%which must be redefined in the context of the finite set of
%machine numbers (see section \ref{Concerning}).

%In addition, the constant of sensitivity will be computed.


%This sensitive dependence could be stated as a consequence of regularity and
%transitivity (by using the theorem of Banks~\cite{Banks92}). However, we
%prefer to prove this result independently of regularity, because the
%notion of regularity must be redefined in the context of the finite set of
%machine numbers (see section \ref{Concerning}).

%In addition, the constant of sensitivity has been computed.

\subsubsection{Devaney's chaos}



In conclusion, $(\mathcal{X}_2,\mathcal{G}_{f_0})$ is topologically transitive, regular,
and has sensitive dependence on initial conditions. Then according to
Devaney's Chaos
defintition~\ref{def:chaos-devaney}~\vpageref{def:chaos-devaney}, we have the result.

\begin{theorem}%[$\mathcal{G}_{f_0}$ is a chaotic map]
\label{theo:chaotic}
$\mathcal{G}_{f_0}$ is a chaotic map on $(\mathcal{X}_2,d_2)$ in the sense of Devaney.
\end{theorem}

So we can claim that:

\begin{theorem}%[Chaos-security of SCISMM]
\label{theo:SCISMM-chaosecurity}
SCISMM is chaos-secure.
\end{theorem}


\subsection{Quantitative property of
$CI_2$}\label{sec:quantitative-properties-ci2}

In this section, it is studied one of the quantitative properties of the
chaos-security of the steganographic process $CI_2$: the constant of sensitivity
evaluation.

\textbf{TODO}

\subsection{Qualitative property of
$CI_2$}\label{sec:qualitative-properties-ci2}

In this section, it is studied one of the qualitative properties of the
chaos-security of the steganographic process $CI_2$: the strong transitivity.

\textbf{TODO}

\subsection{Consequences in attacks
frameworks}\label{sec:qualitative-properties-ci2}

\textbf{TODO} In this section, it is given the implication of the qualitative
and quantitative properties of $CI_2$ in KOA, KMA,CMA and WOA setups.



\section{Conclusion and future work}\label{sec:conclusion}

In this research work, the two information hiding schemes based on chaotic iterations proposed in \cite{gfb10:ip} and \cite{fgb11:ip}
 have been recalled. 
Additionally, all the previous contributions in the field of chaos-based security for information hiding have been summed up. 
Furthermore, the robustness study of the first scheme initiated in \cite{guyeux10ter} has been widely improved, by using a large number of images, parameters, and attacks.
Finally, the security study of the new version of our information hiding scheme has been deepened, by giving qualitative and quantitative measures of its level of chaos-security.
In particular, the constant of sensibility and the qualitative properties of
strong transitivity and indecomposability have been established.\newline

The comparison between the currently only stego and chaos secure schemes is
synthesized in the
table~\ref{table:security-processes-comparison}~\vpageref{table:security-processes-comparison}.\newline

In future work, we intend to compare the robustness and security of these schemes with other existing information hiding algorithms.
Among other things, a complete comparison between spread-spectrum and chaotic iterations will be realized, and the security of other existing schemes will be studied in the framework of chaos-security. 
Concerning the theoretical aspects of security, we intend to understand links and implications between the two existing security frameworks for information hiding: the stego-security and the chaos-security.
Finally, their relation with intended requirements in information hiding will be deepened.


\newcolumntype{V}{>{\centering\arraybackslash} m{1cm} }
%\resizebox{8cm}{!} {
\begin{table}
\begin{center}
%\begin{tabularx}{8cm}{|c|V|V|V|V|}
\begin{tabular}{|c|c|c|c|c|c|c|}\cline{2-3}
  \hline
   & \multirow{2}{*}{\textbf{KOA}} & \multirow{2}{*}{\textbf{KMA}} &
   \multirow{2}{*}{\textbf{WOA}} & \multirow{2}{*}{\textbf{CMA}} &
   \textbf{Bits}\\ & & & & & \textbf{embeded}\\
  \hline
    \hline
  
  \textbf{$CI_1$} &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}} 
   &
  1
  \\
    \hline
  
  \textbf{NW \tiny($\eta=1$)}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-normal-jaune}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-normal-jaune}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/interrogation}}
     &
  N
   \\
      \hline
   
   \hline
   \textbf{$CI_2$} &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
  &
  \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
     &
  N
  \\
   
  \hline
\end{tabular}
\caption{NW, $CI_1$, and $CI_2$ security. Embedding capacity.}
\label{table:security-processes-comparison}
\end{center}

\end{table}


\newpage
%\ack{To do.}

% To cite in reference all the Bibtex entries.
%\nocite{*}

%\bibliographystyle{compj}
%\bibliography{ModellingBidders}

\bibliographystyle{compj}
\bibliography{jabref}


\end{document}
