+\JFC{Dire que c'est une synthèse du chapitre 22 de la thèse de Tof}
+
+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 $x$.
+
+Le processus de marquage est fondé sur les itérations unaires d'une fonction
+selon une stratégie donnée. Cette fonction et cette stratégie
+sont paramétrées par un entier naturel permettant à la méthode d'être
+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]
+\label{def:mode}
+Soit $\mathsf{N}$ un entier naturel.
+Un mode est une application de $\mathds{B}^{\mathsf{N}}$
+dans lui même.
+\end{definition}
+
+
+
+\begin{definition}[Adapteur de Stratégie]
+ \label{def:strategy-adapter}
+
+ Un \emph{adapteur de stratégie} est une fonction $\mathcal{S}$
+ de $\Nats$ dans l'ensemble des séquences d'entiers
+ qui associe à chaque entier naturel
+ $\mathsf{N}$ la suite
+ $S \in \llbracket 1, n\rrbracket^{\mathds{N}}$.
+\end{definition}
+
+
+On définit par exemple l'adapteur CIIS (\emph{Chaotic Iterations with Independent Strategy})
+paramétré par $(K,y,\alpha,l) \in [0,1]\times [0,1] \times ]0, 0.5[ \times \mathds{N}$
+qui associe à chque entier $n \in \Nats$ la suite
+$(S^t)^{t \in \mathds{N}}$ définie par:
+ \begin{itemize}
+ \item $K^0 = \textit{bin}(y) \oplus \textit{bin}(K)$: $K^0$ est le nombre binaire (sur 32 bits)
+ égal au ou exclusif (xor)
+ entre les décompositions binaires sur 32 bits des réels $y$ et $K$
+ (il est aussi compris entre 0 et 1),
+ \item $\forall t \leqslant l, K^{t+1} = F(K^t,\alpha)$,
+ \item $\forall t \leqslant l, S^t = \left \lfloor n \times K^t \right \rfloor + 1$,
+ \item $\forall t > l, S^t = 0$,
+ \end{itemize}
+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 peut attribuer à chaque bit du média hôte $x$ sa valeur d'importance
+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 ]
+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}
+
+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}
+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
+\emph{bits les plus significatifs (MSBs)} de $x$,
+\emph{bits les moins significatifs (LSBs)} de $x$
+\emph{bits passifs} of $x$ définis par:
+\begin{eqnarray*}
+ u_M &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ et } u^k
+ \geqslant M \textrm{ et } k \le \mid x \mid \right) \\
+ u_m &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ et } u^k
+ \le m \textrm{ et } k \le \mid x \mid \right) \\
+ 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}
+
+On peut alors définir une fonction de décompostion
+puis de recomposition pour un hôte $x$:
+
+
+\begin{definition}[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
+$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$
+avec
+\begin{itemize}
+\item $u_M$, $u_m$, et $u_p$ construits comme à la définition~\label{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}
+La fonction qui associe $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$
+pour chaque hôte $x$ est la \emph{fonction de décomposition}, plus tard notée
+$\textit{dec}(u,m,M)$ puisuq'elle est paramétrée par
+$u$, $m$ and $M$.
+\end{definition}
+
+
+\begin{definition}[Recomposition]
+Soit un sextuplet
+$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p}) \in
+\mathfrak{N} \times
+\mathfrak{N} \times
+\mathfrak{N} \times
+\mathfrak{B} \times
+\mathfrak{B} \times
+\mathfrak{B}
+$ tel que
+\begin{itemize}
+\item les ensembles $u_M$, $u_m$ et $u_p$ forment une partition de $[n]$;
+\item $|u_M| = |\varphi_M|$, $|u_m| = |\varphi_m|$ et $|u_p| = |\varphi_p|$.
+\end{itemize}
+Soit la base canonique sur l'espace vectoriel $\mathds{R}^{\mid x \mid}$ composée des vecteurs
+ $e_1, \ldots, e_{\mid x \mid}$.
+On peut construire le vecteur
+\[
+x =
+\sum_{i=1}^{|u_M|} \varphi^i_M . e_{{u^i_M}} +
+\sum_{i=1}^{|u_m|} \varphi^i_m .e_{{u^i_m}} +
+\sum_{i=1}^{|u_p|} \varphi^i_p. e_{{u^i_p}}
+\]
+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}
+
+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]
+Soit une fonction de décomposition $\textit{dec}(u,m,M)$ 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}
+
+Let us then define the dhCI information hiding scheme
+presented in~\cite{gfb10:ip}:
+
+\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.
+
+\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.
+\end{itemize}
+\end{definition}
+
+
+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 $l=|u_m|$.
+
+
+%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.
+
+%\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 $l$-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.
+
+
+
+
+
+
+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{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}
+
+
+Next section shows how to check whether a media contains a mark.
