X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/blobdiff_plain/a1c6f53ee4176cdde6192aeeb03b29aed8549bbd..d1c0e5b780e93cf63c37b5c802d675247a4d5a21:/oxford.tex

diff --git a/oxford.tex b/oxford.tex
index 92d4d92..1f18e05 100644
--- a/oxford.tex
+++ b/oxford.tex
@@ -1,5 +1,8 @@
 \JFC{Dire que c'est une synthÃ¨se du chapitre 22 de la thÃ¨se de Tof}
 
+
+\section{Processus de marquage}
+
 Par la suite, le message numÃ©rique qu'on cherche Ã  embarquer est 
 notÃ© $y$ et le support dans lequel se fait l'insertion est notÃ© $x$. 
 
@@ -9,16 +12,19 @@ sont paramÃ©trÃ©es par un entier naturel permettant Ã  la mÃ©thode d'Ãªtre
 appliquable Ã  un mÃ©dia de n'importe quelle taille.
 On parle alors respectivement de \emph{mode} et d'\emph{adapteur de stratÃ©gies} 
 
-\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 mÃªme.
-\end{definition}
+\end{Def}
 
 
 
-\begin{definition}[Adapteur de StratÃ©gie]
+\begin{Def}[Adapteur de StratÃ©gie]
   \label{def:strategy-adapter}
   
   Un  \emph{adapteur de stratÃ©gie} est une fonction $\mathcal{S}$ 
@@ -26,7 +32,7 @@ dans lui mÃªme.
   qui associe Ã  chaque entier naturel
   $\mathsf{N}$ la suite 
   $S \in  \llbracket 1, n\rrbracket^{\mathds{N}}$.
-\end{definition}
+\end{Def}
 
 
 On dÃ©finit par exemple  l'adapteur CIIS (\emph{Chaotic Iterations with Independent Strategy})
@@ -45,42 +51,7 @@ $(S^t)^{t \in \mathds{N}}$ dÃ©finie par:
 oÃ¹  est la fonction chaotique linÃ©aire par morceau~\cite{Shujun1}.
 Les paramÃ¨tres $K$ et $\alpha$ de cet adapteur de stratÃ©gie peuvent Ãªtre vus
 comme des clefs. 
-
-% Les paramÃ¨re 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 stratÃ©gie est unaire.
 
 
 
@@ -89,27 +60,21 @@ On peut attribuer Ã  chaque bit du mÃ©dia hÃ´te $x$ sa valeur d'importance
 sous la forme d'un rÃ©el.
 Ceci se fait Ã  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 
 sÃ©quence finie de bit $x$ associe la sÃ©quence 
 $(u^k(x))$ de taille $\mid x \mid$ Ã  valeur dans les rÃ©els.
 Cette fonction peut dÃ©pendre du message $y$ Ã  embarquer, ou non.
-\end{definition}
+\end{Def}
 
 Pour allÃ©ger le discours, par la suite, on nottera $(u^k(x))$ pour $(u^k)$ 
 lorsque cela n'est pas ambigÃ¼e.
 Il reste Ã  partionner les bits  de $x$ selon qu'ils sont 
 peu, moyennement ou trÃ¨s 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 rÃ©els  t.q. $m < M$.  Alors:
 $u_M$, $u_m$ et $u_p$ sont les vecteurs finis respectivements des 
@@ -124,13 +89,13 @@ $u_M$, $u_m$ 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 dÃ©finir une fonction de dÃ©compostion  
 puis de recomposition pour un hÃ´te $x$:
 
 
-\begin{definition}[Fonction de dÃ©composition ]
+\begin{Def}[Fonction de dÃ©composition ]
 Soit $u$ une fonction de signification, 
 $m$ et $M$ deux rÃ©els t.q  $m < M$.  
 Tout hÃ´te $x$ peut se dÃ©composer en
@@ -144,12 +109,12 @@ avec
 \end{itemize}
 La fonction qui associe $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$
 pour chaque hÃ´te $x$ est la  \emph{fonction de dÃ©composition}, plus tard notÃ©e 
-$\textit{dec}(u,m,M)$ puisuq'elle est paramÃ©trÃ©e par 
+$\textit{dec}(u,m,M)$ puisqu'elle est paramÃ©trÃ©e 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 
@@ -175,241 +140,354 @@ x =
 La fonction qui associe $x$ Ã  chaque sextuplet 
 $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ dÃ©fini comme ci-dessus est appelÃ©e 
 \emph{fonction de recomposition}.
-\end{definition}
+\end{Def}
 
 Un embarquement consiste Ã  modifier les valeurs de  
 $\phi_{m}$ (de $x$) en tenant compte de $y$. 
 Cela se formalise comme suit:
 
-\begin{definition}[Embedding media]
-Soit une fonction de dÃ©composition  $\textit{dec}(u,m,M)$ 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 $\textit{rec}$.
+\begin{Def}[Embarquement de message]
+Soit une fonction de dÃ©composition  $\textit{dec}(u,m,M)$,  
+$x$ 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 mÃ©dia numÃ©rique de taille $|u_m|$.
+Le mÃ©dia $z$ rÃ©sultant de l'embarquement d'$y$ dans $x$ est l'image de 
+$(u_M,u_m,u_p,\phi_{M},y,\phi_{p})$
+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}
+
+ 
+On peut Ã©tendre l'algorithme dhCI~\cite{gfb10:ip} d'embarquement de message comme suit:  
+
+\begin{Def}[Embarquement dhCI Ã©tendu]
+ \label{def:dhCI:ext}
+Soit $\textit{dec}(u,m,M)$ une function de dÃ©composition,
+$f$ un mode, 
+$\mathcal{S}$ un adapteur de stratÃ©gie,
+$x$ un hÃ´te, 
+$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ 
+sont image par  $\textit{dec}(u,m,M)$,
+$q$ un entier naturel positif
+et $y$ un mÃ©dia numÃ©rique de taille $l=|u_m|$.
+
+L'algorithme d'embarquement de message associe Ã  chaque 
+couple $(x,y)$  le mÃ©dia  $z$ rÃ©sultat de l'embarquement de 
+$\hat{y}$ dans $x$, t. q.:
+
+\begin{itemize}
+\item le mode $f$ est instanciÃ© avec le paramÃ¨tre $l=|u_m|$, engendrant la 
+  fonction $f_{l}:\Bool^{l} \rightarrow \Bool^{l}$;
+\item l'adapteur de stratÃ©gie $\mathcal{S}$ est intanciÃ© avec le paramÃ¨tre
+$y$, engendrant une stratÃ©gie $S_y \in [l]$;
+\item on itÃ¨re la fonction $G_{f_l}$ en prenant la configuration
+  initiale $(S_y,\phi_{m})$ selon le schÃ©ma dÃ©fini 
+  Ã  l'Ã©quation~(\ref{eq:sch:unaire}). 
+\item $\hat{y}$ est le second membre du $q^{\textrm{Ã¨me}}$ terme obtenu.
+\end{itemize}
+\end{Def}
+
+
+La figure~\ref{fig:organigramme} synthÃ©tise la dÃ©marche.
+
+\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{DÃ©tection d'un media marquÃ©}\label{sub:wmdecoding}
+
+On caractÃ©rise d'abord ce qu'est un mÃ©dia marquÃ© selon la mÃ©thode Ã©noncÃ©e 
+Ã  la section prÃ©cÃ©dente. On considÃ¨re que l'on connaÃ®t
+la marque Ã  embarquer $y$, le support $x$ et que l'on a face Ã  soit un mÃ©dia 
+$z$.
+
+
+\begin{definition}[MÃ©dia marquÃ©]
+Soit $\textit{dec}(u,m,M)$ une fonction de dÃ©composition
+$f$ un  mode, 
+$\mathcal{S}$ un adapteur de stratÃ©gie
+$q$ un entier naturel strictement positif,
+$y$ un mÃ©dia digital et soit  
+$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ l'image par 
+$\textit{dec}(u,m,M)$  du mÃ©dia  $x$. 
+Alors, $z$ est \emph{marquÃ©} avec $y$ si l'image 
+par $\textit{dec}(u,m,M)$ of $z$ is 
+$(u_M,u_m,u_p,\phi_{M},\hat{y},\phi_{p})$, oÃ¹
+$\hat{y}$ est le second membre de  $G_{f_l}^q(S_y,\phi_{m})$.
 \end{definition}
 
-Let us then define the dhCI information hiding scheme
-presented in~\cite{gfb10:ip}:
+% Plusieurs stratÃ©gies peuvent Ãªtre utilisÃ©es pour dÃ©terminer si une image $z$ 
+% est marquÃ©e, en particulier si l'image a Ã©tÃ© attaquÃ©e entre temps.
+% On s'intÃ©ressera aux mesures de similaritÃ© entre $x$ et $z$.
+
+\section{Analyse de sÃ©curitÃ©}\label{sec:security}
 
-\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 $\textit{dec}(u,m,M)$,
-$q$ be a positive natural number,  
-and $y$ be a digital media of size $l=|u_m|$.
 
 
-The dhCI dissimulation  maps any
-$(x,y)$  to the digital media $z$ resulting on the embedding of
-$\hat{y}$ into $x$, s.t.
 
+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. 
+
+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 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 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}
-\end{definition}
 
+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.
 
-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}
 
-%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 $l=|u_m|$.
+\subsection{Stego-security}\label{sub:stegosecurity}
+%\input{stegosecurity}
 
 
-%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.
+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.
 
-%\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 $S_y \in \llbracket 1;l\rrbracket ^{\Nats}$.
+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}.
 
-%\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}
 
+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.
 
-%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.
+\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.
 
-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.
+%\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}
 
-\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}
+ 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 lÃ  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}
