\JFC{Dire que c'est une synthèse du chapitre 22 de la thèse de Tof}
+
+\section{Processus de marquage}
+
Par la suite, le message numérique qu'on cherche à embarquer est
noté $y$ et le support dans lequel se fait l'insertion est noté $x$.
appliquable à un média de n'importe quelle taille.
On parle alors respectivement de \emph{mode} et d'\emph{adapteur de stratégies}
-\begin{definition}[Mode]
+\subsection{Embarquement}
+
+
+\begin{Def}[Mode]
\label{def:mode}
Soit $\mathsf{N}$ un entier naturel.
Un mode est une application de $\mathds{B}^{\mathsf{N}}$
dans lui même.
-\end{definition}
+\end{Def}
-\begin{definition}[Adapteur de Stratégie]
+\begin{Def}[Adapteur de Stratégie]
\label{def:strategy-adapter}
Un \emph{adapteur de stratégie} est une fonction $\mathcal{S}$
qui associe à chaque entier naturel
$\mathsf{N}$ la suite
$S \in \llbracket 1, n\rrbracket^{\mathds{N}}$.
-\end{definition}
+\end{Def}
On définit par exemple l'adapteur CIIS (\emph{Chaotic Iterations with Independent Strategy})
où est la fonction chaotique linéaire par morceau~\cite{Shujun1}.
Les paramètres $K$ et $\alpha$ de cet adapteur de stratégie peuvent être vus
comme des clefs.
-
-% Les paramère Parameters of CIIS strategy-adapter will be instantiate as follows:
-% $K$ is the secret embedding key, $y$ is the secret message,
-% $\alpha$ is the threshold of the piecewise linear chaotic map,
-% which can be set as $K$ or can act as a second secret key.
-% Lastly, $l$ is for the iteration number bound:
-% enlarging its value improve the chaotic behavior of the scheme,
-% but the time required to achieve the embedding grows too.
-
-% Another strategy-adapter is the
-% \emph{Chaotic Iterations with Dependent Strategy} (CIDS)
-% with parameters $(l,X) \in \mathds{N}\times \mathds{B}^\mathds{N}$,
-% which is the function that maps any $ n \in \mathds{N}$ to
-% the sequence $\left(S^t\right)^{t \in \mathds{N}}$ defined by:
-% \begin{itemize}
-% \item $\forall t \leqslant l$, if $t \leqslant l$ and $X^t = 1$,
-% then $S^t=t$, else $S^t=1$.
-% \item $\forall t > l, S^t = 0$.
-% \end{itemize}
-
-
-
-
-% Let us notice that the terms of $x$ that may be replaced by terms issued
-% from $y$ 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,
-% w.r.t. its position $t$:
-
-% \begin{definition}[Signification function]
-% A \emph{signification function}
-% $(u^k)^{k \in \Nats}$. % with a limit equal to 0.
-% \end{definition}
-
-
+On remarque que cette stratégie est unaire.
sous la forme d'un réel.
Ceci se fait à l'aide d'une fonction de signification.
-%We first notice that terms of $x$ that may be replaced by terms issued
-%from $y$ are less important than other: they could be changed
-%without being perceived as such. More generally, a
-%\emph{signification function}
-%attaches a weight to each terms defining a digital media,
-%depending on its position $t$, as follows.
-\begin{definition}[Fonction de signification ]
+\begin{Def}[Fonction de signification ]
Une \emph{fonction de signification }
est une fonction $u$ qui a toute
séquence finie de bit $x$ associe la séquence
$(u^k(x))$ de taille $\mid x \mid$ à valeur dans les réels.
Cette fonction peut dépendre du message $y$ à embarquer, ou non.
-\end{definition}
+\end{Def}
Pour alléger le discours, par la suite, on nottera $(u^k(x))$ pour $(u^k)$
lorsque cela n'est pas ambigüe.
Il reste à partionner les bits de $x$ selon qu'ils sont
peu, moyennement ou très significatifs.
-\begin{definition}[Signification des bits]\label{def:msc,lsc}
+\begin{Def}[Signification des bits]\label{def:msc,lsc}
Soit $u$ une fonction de signification,
$m$ et $M$ deux réels t.q. $m < M$. Alors:
$u_M$, $u_m$ et $u_p$ sont les vecteurs finis respectivements des
u_p &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ et }
u^k \in ]m;M[ \textrm{ et } k \le \mid x \mid \right)
\end{eqnarray*}
- \end{definition}
+ \end{Def}
On peut alors définir une fonction de décompostion
puis de recomposition pour un hôte $x$:
-\begin{definition}[Fonction de décomposition ]
+\begin{Def}[Fonction de décomposition ]
Soit $u$ une fonction de signification,
$m$ et $M$ deux réels t.q $m < M$.
Tout hôte $x$ peut se décomposer en
pour chaque hôte $x$ est la \emph{fonction de décomposition}, plus tard notée
$\textit{dec}(u,m,M)$ puisqu'elle est paramétrée par
$u$, $m$ and $M$.
-\end{definition}
+\end{Def}
-\begin{definition}[Recomposition]
+\begin{Def}[Recomposition]
Soit un sextuplet
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p}) \in
\mathfrak{N} \times
La fonction qui associe $x$ à chaque sextuplet
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ défini comme ci-dessus est appelée
\emph{fonction de recomposition}.
-\end{definition}
+\end{Def}
Un embarquement consiste à modifier les valeurs de
$\phi_{m}$ (de $x$) en tenant compte de $y$.
Cela se formalise comme suit:
-\begin{definition}[Embedding media]
+\begin{Def}[Embarquement de message]
Soit une fonction de décomposition $\textit{dec}(u,m,M)$,
$x$ un support,
$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ son image par $\textit{dec}(u,m,M)$,
et $y$ un média numérique de taille $|u_m|$.
Le média $z$ résultant de l'embarquement d'$y$ dans $x$ est l'image de
-$(u_M,u_m,u_p,\phi_{M},g(\phi_{m},y),\phi_{p})$
-par la fonction de recomposition $\textit{rec}$ avec
-$g : \Bool^{|u_m|} \times \Bool^{|u_m|} \to \Bool^{|u_m|} $
-est la fonction de modification des bits de $u_m$ selon $y$.
-\end{definition}
+$(u_M,u_m,u_p,\phi_{M},y,\phi_{p})$
+par la fonction de recomposition $\textit{rec}$.
+% avec
+% $g : \Bool^{|u_m|} \times \Bool^{|u_m|} \to \Bool^{|u_m|} $
+% est la fonction de modification des bits de $u_m$ selon $y$.
+\end{Def}
-Dans l'embaquement LSB,
-$u$ est la fonction qui asocie 0 aux bits de poids faible de chaque pixel et 1 ailleur,
-$m$ et $M$ valent respectivement 0 et 1 et
-$g$ remplace (\teixtit{i.e.}, écrase) tous les bits de $u_m$ par ceux de $y$.
On peut étendre l'algorithme dhCI~\cite{gfb10:ip} d'embarquement de message comme suit:
-\begin{definition}[]
+\begin{Def}[Embarquement dhCI étendu]
\label{def:dhCI:ext}
Soit $\textit{dec}(u,m,M)$ une function de décomposition,
$f$ un mode,
L'algorithme d'embarquement de message associe à chaque
couple $(x,y)$ le média $z$ résultat de l'embarquement de
-$\hat{y}$ dans $x$, t. q.
-%%%%%%%%%%%%%%%%%%%%%%%%
+$\hat{y}$ dans $x$, t. q.:
+
\begin{itemize}
-\item We instantiate the mode $f$ with parameter $l=|u_m|$, leading to
- the function $f_{l}:\Bool^{l} \rightarrow \Bool^{l}$.
-\item We instantiate the strategy adapter $\mathcal{S}$
-with parameter $y$ (and some other ones eventually).
-This instantiation leads to the strategy $S_y \in \llbracket 1;l\rrbracket ^{\Nats}$.
-
-\item We iterate $G_{f_l}$ with initial configuration $(S_y,\phi_{m})$.
-\item $\hat{y}$ is finally the $q$-th term of these iterations.
+\item le mode $f$ est instancié avec le paramètre $l=|u_m|$, engendrant la
+ fonction $f_{l}:\Bool^{l} \rightarrow \Bool^{l}$;
+\item l'adapteur de stratégie $\mathcal{S}$ est intancié avec le paramètre
+$y$, engendrant une stratégie $S_y \in [l]$;
+\item on itère la fonction $G_{f_l}$ en prenant la configuration
+ initiale $(S_y,\phi_{m})$ selon le schéma défini
+ à l'équation~(\ref{eq:sch:unaire}).
+\item $\hat{y}$ est le second membre du $q^{\textrm{ème}}$ terme obtenu.
\end{itemize}
+\end{Def}
+
+
+La figure~\ref{fig:organigramme} synthétise la démarche.
+
+\begin{figure}[ht]
+\centering
+%\includegraphics[width=8.5cm]{organigramme2.pdf}
+\includegraphics[width=8.5cm]{organigramme22}
+\caption{The dhCI dissimulation scheme}
+\label{fig:organigramme}
+\end{figure}
+
+
+
+
+\subsection{Détection d'un media marqué}\label{sub:wmdecoding}
+
+On caractérise d'abord ce qu'est un média marqué selon la méthode énoncée
+à la section précédente. On considère que l'on connaît
+la marque à embarquer $y$, le support $x$ et que l'on a face à soit un média
+$z$.
+
+
+\begin{definition}[Média marqué]
+Soit $\textit{dec}(u,m,M)$ une fonction de décomposition
+$f$ un mode,
+$\mathcal{S}$ un adapteur de stratégie
+$q$ un entier naturel strictement positif,
+$y$ un média digital et soit
+$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ l'image par
+$\textit{dec}(u,m,M)$ du média $x$.
+Alors, $z$ est \emph{marqué} avec $y$ si l'image
+par $\textit{dec}(u,m,M)$ of $z$ is
+$(u_M,u_m,u_p,\phi_{M},\hat{y},\phi_{p})$, où
+$\hat{y}$ est le second membre de $G_{f_l}^q(S_y,\phi_{m})$.
\end{definition}
+% Plusieurs stratégies peuvent être utilisées pour déterminer si une image $z$
+% est marquée, en particulier si l'image a été attaquée entre temps.
+% On s'intéressera aux mesures de similarité entre $x$ et $z$.
-To summarize, iterations are realized on the LSCs of the
-host content
-(the mode gives the iterate function,
-the strategy-adapter gives its strategy),
-and the last computed configuration is re-injected into the host content,
-in place of the former LSCs.
-
-
-
-%\begin{definition}[Significance of coefficients]\label{def:msc,lsc}
-%Let $(u^k)^{k \in \Nats}$ be a signification function,
-%$m$ and $M$ be two reals s.t. $m < M$. Then
-%the \emph{most significant coefficients (MSCs)} of $x$ is the finite
-% vector $u_M$,
-%the \emph{least significant coefficients (LSCs)} of $x$ is the
-%finite vector $u_m$, and
-%the \emph{passive coefficients} of $x$ is the finite vector $u_p$ such that:
-%\begin{eqnarray*}
-% u_M &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k
-% \geqslant M \textrm{ and } k \le \mid x \mid \right) \\
-% u_m &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k
-% \le m \textrm{ and } k \le \mid x \mid \right) \\
-% 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{eqnarray*}
-% \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.
-%We are then ready to decompose an host $x$ into its coefficients and
-%then to recompose it. Next definitions formalize these two steps.
-
-%\begin{definition}[Decomposition function]
-%Let $(u^k)^{k \in \Nats}$ be a signification function,
-%$\mathfrak{B}$ the set of finite binary sequences,
-%$\mathfrak{N}$ the set of finite integer sequences,
-%$m$ and $M$ be two reals s.t. $m < M$.
-%Any host $x$ may be decomposed into
-%$$
-%(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})
-%\in
-%\mathfrak{N} \times
-%\mathfrak{N} \times
-%\mathfrak{N} \times
-%\mathfrak{B} \times
-%\mathfrak{B} \times
-%\mathfrak{B}
-%$$
-%where
-%\begin{itemize}
-%\item $u_M$, $u_m$, and $u_p$ are coefficients defined in Definition
-%\ref{def:msc,lsc};
-%\item $\phi_{M} = \left( x^{u^1_M}, x^{u^2_M}, \ldots,x^{u^{|u_M|}_M}\right)$;
-% \item $\phi_{m} = \left( x^{u^1_m}, x^{u^2_m}, \ldots,x^{u^{|u_m|}_m} \right)$;
-% \item $\phi_{p} =\left( x^{u^1_p}, x^{u^2_p}, \ldots,x^{u^{|u_p|}_p}\right) $.
-% \end{itemize}
-%The function that associates the decomposed host to any digital host is
-%the \emph{decomposition function}. It is
-%further referred as $\textit{dec}(u,m,M)$ since it is parametrized by
-%$u$, $m$, and $M$. Notice that $u$ is a shortcut for $(u^k)^{k \in \Nats}$.
-%\end{definition}
-
-
-%\begin{definition}[Recomposition]
-%Let
-%$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p}) \in
-%\mathfrak{N} \times
-%\mathfrak{N} \times
-%\mathfrak{N} \times
-%\mathfrak{B} \times
-%\mathfrak{B} \times
-%\mathfrak{B}
-%$ s.t.
-%\begin{itemize}
-%\item the sets of elements in $u_M$, elements in $u_m$, and
-%elements in $u_p$ are a partition of $\llbracket 1, n\rrbracket$;
-%\item $|u_M| = |\varphi_M|$, $|u_m| = |\varphi_m|$, and $|u_p| = |\varphi_p|$.
-%\end{itemize}
-%One may associate the vector
-%$$x =
-%\sum_{i=1}^{|u_M|} \varphi^i_M . e_{{u^i_M}} +
-%\sum_{i=1}^{|u_m|} \varphi^i_m .e_{{u^i_m}} +
-%\sum_{i=1}^{|u_p|} \varphi^i_p. e_{{u^i_p}}
-%$$
-%\noindent where $e_i$ is the sequence whose $j-$th term is equal to $\overline{\Delta(i,j)}$, \emph{i.e.}, $(e_i)_{i \in \mathds{N}}$ is the usual basis of the $\mathds{R}-$vectorial space $\left(\mathds{R}^\mathds{N}, +, .\right)$.
-%The function that associates $x$ to any
-%$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ following the above constraints
-%is called the \emph{recomposition function}.
-%\end{definition}
+\section{Analyse de sécurité}\label{sec:security}
-%The embedding consists to the replacement of the values of
-%$\phi_{m}$ of $x$'s LSCs by $y$.
-%It then composes the two decomposition and
-%recomposition functions seen previously. More formally:
-
-
-%\begin{definition}[Embedding digital media]
-%Let $\textit{dec}(u,m,M)$ be a decomposition function,
-%$x$ be a host content,
-%$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be its image by $\textit{dec}(u,m,M)$,
-%and $y$ be a digital media of size $|u_m|$.
-%The digital media $z$ resulting on the embedding of $y$ into $x$ is
-%% the
-%% result of the \emph{embedding} of $y$ in $x$ if
-%% $$
-%% \forall n \in \llbracket1, |x|\rrbracket , z^n = \left\{
-%% \begin{array}{ll}
-%% x^n & \textrm{if } \phi^n_m > m,\\
-%% y^n & \textrm{else.}
-%% \end{array}
-%% \right.
-%% $$
-%%
-%% In other words, $z$ is
-%the image of $(u_M,u_m,u_p,\phi_{M},y,\phi_{p})$
-%by the recomposition function $\textit{rec}$.
-%\end{definition}
-%We can now define the information hiding scheme called \emph{dhCI}:
-%\begin{definition}[Data hiding dhCI]
-% \label{def:dhCI}
-%Let $\textit{dec}(u,m,M)$ be a decomposition function,
-%$f$ be a mode,
-%$\mathcal{S}$ be a strategy adapter,
-%$x$ be an host content,\linebreak
-%$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be its image by $\textit{dec}(u,m,M)$,
-%and $y$ be a digital media of size $l=|u_m|$.
+As far as we know, Cachin~\cite{Cachin2004}
+produces the first fundamental work in information hiding security:
+in the context of steganography, the attempt of an attacker to distinguish
+between an innocent image and a stego-content is viewed as an hypothesis
+testing problem.
+Mittelholzer~\cite{Mittelholzer99} next proposed the first theoretical
+framework for analyzing the security of a watermarking scheme.
+Clarification between robustness and security
+and classifications of watermarking attacks
+have been firstly presented by Kalker~\cite{Kalker2001}.
+This work has been deepened by Furon \emph{et al.}~\cite{Furon2002}, who have translated Kerckhoffs' principle (Alice and Bob shall only rely on some previously shared secret for privacy), from cryptography to data hiding.
-%The \emph{dhCI dissimulation} is the application that maps any
-%$(x,y)$ to the digital media $z$ resulting on the embedding of
-%$\hat{y}$ into $x$, s.t.
+More recently~\cite{Cayre2005,Perez06} classified the information hiding
+attacks into categories, according to the type of information the attacker (Eve)
+has access to:
+\begin{itemize}
+\item in Watermarked Only Attack (WOA) she only knows embedded contents $z$;
+\item in Known Message Attack (KMA) she knows pairs $(z,y)$ of embedded
+ contents and corresponding messages;
+\item in Known Original Attack (KOA) she knows several pairs $(z,x)$
+ of embedded contents and their corresponding original versions;
+\item in Constant-Message Attack (CMA) she observes several embedded
+ contents $z^1$,\ldots,$z^k$ and only knows that the unknown
+ hidden message $y$ is the same in all contents.
+\end{itemize}
-%\begin{itemize}
-%\item We instantiate the mode $f$ with parameter $l=|u_m|$, leading to
-% the function $f_{l}:\Bool^{l} \rightarrow \Bool^{l}$.
-%\item We instantiate the strategy adapter $\mathcal{S}$
-%with parameter $y$ (and some other ones eventually).
-%This instantiation leads to the strategy $S_y \in \llbracket 1;l\rrbracket ^{\Nats}$.
+To the best of our knowledge,
+KMA, KOA, and CMA have not already been studied
+due to the lack of theoretical framework.
+In the opposite, security of data hiding against WOA can be evaluated,
+by using a probabilistic approach recalled below.
-%\item We iterate $G_{f_l}$ with initial configuration $(S_y,\phi_{m})$.
-%\item $\hat{y}$ is finally the $l$-th term of these iterations.
-%\end{itemize}
-%\end{definition}
-%To summarize, some iterations are realized on the LSCs of the
-%host content
-%(the mode gives the iterate function,
-%the strategy-adapter gives its strategy),
-%and the last computed state is re-injected into the host content,
-%in place of the former LSCs.
+\subsection{Stego-security}\label{sub:stegosecurity}
+%\input{stegosecurity}
+In the Simmons' prisoner problem~\cite{Simmons83}, 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 named Eve, who eavesdrops all contents and can choose
+to interrupt the communication if they appear to be stego-contents.
+Stego-security, defined in this well-known context, is the highest security
+class in Watermark-Only Attack setup, which occurs when Eve has only access to
+several marked contents~\cite{Cayre2008}.
-Notice that in order to preserve the unpredictable behavior of the system,
-the size of the digital medias is not fixed.
-This approach is thus self adapted to any media, and more particularly to
-any size of LSCs.
-However this flexibility enlarges the complexity of the presentation:
-we had to give Definitions~\ref{def:mode} and~\ref{def:strategy-adapter}
-respectively of mode and strategy adapter.
+Let $\mathds{K}$ be the set of embedding keys, $p(X)$ the probabilistic model of
+$N_0$ initial host contents, and $p(Y|K)$ the probabilistic model of $N_0$
+marked contents s.t. each host content has been marked
+with the same key $K$ and the same embedding function.
-\begin{figure}[ht]
-\centering
-%\includegraphics[width=8.5cm]{organigramme2.pdf}
-\includegraphics[width=8.5cm]{organigramme2.eps}
-\caption{The dhCI dissimulation scheme}
-\label{fig:organigramme}
-\end{figure}
+\begin{definition}[Stego-Security~\cite{Cayre2008}]
+\label{Def:Stego-security} The embedding function is \emph{stego-secure}
+if $\forall K \in \mathds{K}, p(Y|K)=p(X)$ is established.
+\end{definition}
+
+
+
+
+
+%Let $\mathds{K}$ be the set of embedding keys, $p(X)$ the probabilistic model of
+%$N_0$ initial host contents, and $p(Y|K)$ the probabilistic model of $N_0$
+%marked contents s.t. each host content has been marked
+%with the same key $K$ and the same embedding function.
+
+%\begin{definition}[Stego-Security]
+%\label{Def:Stego-security} The embedding function is \emph{stego-secure}
+%if $\forall K \in \mathds{K}, p(Y|K)=p(X)$ is established.
+%\end{definition}
+ Stego-security states that the knowledge of $K$ does not help to make the
+ difference between $p(X)$ and $p(Y)$. This definition implies the following
+ property:
+ $$p(Y|K_1)= \cdots = p(Y|K_{N_k})=p(Y)=p(X)$$
+ This property is equivalent to a zero Kullback-Leibler divergence, which is the
+ accepted definition of the "perfect secrecy" in steganography~\cite{Cachin2004}.
+
+
+\subsection{The negation mode is stego-secure}
+To make this article self-contained, this section recalls theorems and proofs of stego-security for negation mode published in~\cite{gfb10:ip}.
+
+\begin{proposition} \emph{dhCI dissimulation} of Definition \ref{def:dhCI} with
+negation mode and CIIS strategy-adapter is stego-secure, whereas it is not the
+case when using CIDS strategy-adapter.
+\end{proposition}
+
+
+\begin{proof} On the one hand, let us suppose that $X \sim
+\mathbf{U}\left(\mathbb{B}^n\right)$ when using \linebreak CIIS$(K,\_,\_,l)$.
+We prove by a
+mathematical induction that $\forall t \in \mathds{N}, X^t \sim
+\mathbf{U}\left(\mathbb{B}^n\right)$.
+
+The base case is immediate, as $X^0 = X \sim
+\mathbf{U}\left(\mathbb{B}^n\right)$. Let us now suppose that the statement $X^t
+\sim \mathbf{U}\left(\mathbb{B}^n\right)$ holds until for some $t$.
+Let $e \in
+\mathbb{B}^n$ and \linebreak $\mathbf{B}_k=(0,\cdots,0,1,0,\cdots,0) \in
+\mathbb{B}^n$ (the digit $1$ is in position $k$).
+
+So
+$P\left(X^{t+1}=e\right)=\sum_{k=1}^n
+P\left(X^t=e\oplus\mathbf{B}_k,S^t=k\right)$ where
+$\oplus$ is again the bitwise exclusive or.
+These two events are independent when
+using CIIS strategy-adapter
+(contrary to CIDS, CIIS is not built by using $X$),
+ thus:
+$$P\left(X^{t+1}=e\right)=\sum_{k=1}^n
+P\left(X^t=e\oplus\mathbf{B}_k\right) \times P\left(S^t=k\right).$$
+
+According to the
+inductive hypothesis: $P\left(X^{n+1}=e\right)=\frac{1}{2^n} \sum_{k=1}^n
+P\left(S^t=k\right)$. The set of events $\left \{ S^t=k \right \}$ for $k \in
+\llbracket 1;n \rrbracket$ is a partition of the universe of possible, so
+$\sum_{k=1}^n P\left(S^t=k\right)=1$. Finally,
+$P\left(X^{t+1}=e\right)=\frac{1}{2^n}$, which leads to $X^{t+1} \sim
+\mathbf{U}\left(\mathbb{B}^n\right)$. This result is true for all $t \in
+\mathds{N}$ and then for $t=l$.
+
+Since $P(Y|K)$ is $P(X^l)$ that is proven to be equal to $P(X)$,
+we thus have established that,
+$$\forall K \in [0;1], P(Y|K)=P(X^{l})=P(X).$$
+So dhCI dissimulation with CIIS
+strategy-adapter is stego-secure.
+
+On the other hand, due to the definition of CIDS, we have \linebreak
+$P(Y=(1,1,\cdots,1)|K)=0$.
+%\JFC{Pourquoi? Justifier davantage là ou dans la def de CIDS}
+So there is no uniform repartition for the stego-contents $Y|K$.
+\end{proof}
+
+
+
+To sum up, Alice and Bob can counteract Eve's attacks in WOA setup, when using
+dhCI dissimulation with CIIS strategy-adapter. To our best knowledge, this is
+the second time an information hiding scheme has been proven to be stego-secure:
+the former was the spread-spectrum technique in natural marking
+configuration with $\eta$ parameter equal to 1 \cite{Cayre2008}.
+
+
+
+
+
+\subsection{A new class of $\varepsilon$-stego-secure schemes}
+
+Let us prove that,
+\begin{theorem}\label{th:stego}
+Let $\epsilon$ be positive,
+$l$ be any size of LSCs,
+$X \sim \mathbf{U}\left(\mathbb{B}^l\right)$,
+$f_l$ be an image mode s.t.
+$\Gamma(f_l)$ is strongly connected and
+the Markov matrix associated to $f_l$
+is doubly stochastic.
+In the instantiated \emph{dhCI dissimulation} algorithm
+with any uniformly distributed (u.d.) strategy-adapter
+that is independent from $X$,
+there exists some positive natural number $q$ s.t.
+$|p(X^q)- p(X)| < \epsilon$.
+\end{theorem}
+
+
+\begin{proof}
+Let $\textit{deci}$ be the bijection between $\Bool^{l}$ and
+$\llbracket 0, 2^l-1 \rrbracket$ that associates the decimal value
+of any binary number in $\Bool^{l}$.
+The probability $p(X^t) = (p(X^t= e_0),\dots,p(X^t= e_{2^l-1}))$ for $e_j \in \Bool^{l}$ is thus equal to
+$(p(\textit{deci}(X^t)= 0,\dots,p(\textit{deci}(X^t)= 2^l-1))$ further denoted by $\pi^t$.
+Let $i \in \llbracket 0, 2^l -1 \rrbracket$,
+the probability $p(\textit{deci}(X^{t+1})= i)$ is
+\[
+ \sum\limits^{2^l-1}_{j=0}
+\sum\limits^{l}_{k=1}
+p(\textit{deci}(X^{t}) = j , S^t = k , i =_k j , f_k(j) = i_k )
+\]
+\noindent
+where $ i =_k j $ is true iff the binary representations of
+$i$ and $j$ may only differ for the $k$-th element,
+and where
+$i_k$ abusively denotes, in this proof, the $k$-th element of the binary representation of
+$i$.
+
+Next, due to the proposition's hypotheses on the strategy,
+$p(\textit{deci}(X^t) = j , S^t = k , i =_k j, f_k(j) = i_k )$ is equal to
+$\frac{1}{l}.p(\textit{deci}(X^t) = j , i =_k j, f_k(j) = i_k)$.
+Finally, since $i =_k j$ and $f_k(j) = i_k$ are constant during the
+iterative process and thus does not depend on $X^t$, we have
+\[
+\pi^{t+1}_i = \sum\limits^{2^l-1}_{j=0}
+\pi^t_j.\frac{1}{l}
+\sum\limits^{l}_{k=1}
+p(i =_k j, f_k(j) = i_k ).
+\]
+
+Since
+$\frac{1}{l}
+\sum\limits^{l}_{k=1}
+p(i =_k j, f_k(j) = i_k )
+$ is equal to $M_{ji}$ where $M$ is the Markov matrix associated to
+ $f_l$ we thus have
+\[
+\pi^{t+1}_i = \sum\limits^{2^l-1}_{j=0}
+\pi^t_j. M_{ji} \textrm{ and thus }
+\pi^{t+1} = \pi^{t} M.
+\]
-Next section shows how to check whether a media contains a mark.
+% The calculus of $p(X^{t+1} = e)$ is thus equal to
+% $\pi^{t+1}_i$.
+
+First of all,
+since the graph $\Gamma(f)$ is strongly connected,
+then for all vertices $i$ and $j$, a path can
+be found to reach $j$ from $i$ in at most $2^l$ steps.
+There exists thus $k_{ij} \in \llbracket 1, 2^l \rrbracket$ s.t.
+${M}_{ij}^{k_{ij}}>0$.
+As all the multiples $l \times k_{ij}$ of $k_{ij}$ are such that
+${M}_{ij}^{l\times k_{ij}}>0$,
+we can conclude that, if
+$k$ is the least common multiple of $\{k_{ij} \big/ i,j \in \llbracket 1, 2^l \rrbracket \}$ thus
+$\forall i,j \in \llbracket 1, 2^l \rrbracket, {M}_{ij}^{k}>0$ and thus
+$M$ is a regular stochastic matrix.
+
+
+Let us now recall the following stochastic matrix theorem:
+\begin{theorem}[Stochastic Matrix]
+ If $M$ is a regular stochastic matrix, then $M$
+ has an unique stationary probability vector $\pi$. Moreover,
+ if $\pi^0$ is any initial probability vector and
+ $\pi^{t+1} = \pi^t.M $ for $t = 0, 1,\dots$ then the Markov chain $\pi^t$
+ converges to $\pi$ as $t$ tends to infinity.
+\end{theorem}
+
+Thanks to this theorem, $M$
+has an unique stationary probability vector $\pi$.
+By hypothesis, since $M$ is doubly stochastic we have
+$(\frac{1}{2^l},\dots,\frac{1}{2^l}) = (\frac{1}{2^l},\dots,\frac{1}{2^l})M$
+and thus $\pi = (\frac{1}{2^l},\dots,\frac{1}{2^l})$.
+Due to the matrix theorem, there exists some
+$q$ s.t.
+$|\pi^q- \pi| < \epsilon$
+and the proof is established.
+Since $p(Y| K)$ is $p(X^q)$ the method is then $\epsilon$-stego-secure
+provided the strategy-adapter is uniformly distributed.
+ \end{proof}
+
+This section has focused on security with regards to probabilistic behaviors.
+Next section studies it in the perspective of topological ones.
+
+
+
+%\subsection{Security in KMA, KOA and CMA setups}
+%\input{KMOA.tex}
