% \subsection{Recall about \CID and topology}\label{sec:recall-ci2-topo}


% \subsubsection{Notations and
% terminologies}\label{sec:ci2-topological-model-notations}



\subsection{Topological Model}
\label{sec:cis2-topological-model}

% \section{THE NEW TOPOLOGICAL SPACE}
In this section, we firstly recall the topology used in~\cite{fgb11:ip}, 
to model the steganographic scheme \CID as a
discrete dynamical system in a topological space. 
The security level of \CID is then recalled
at the end of this section.




% 
% \begin{definition}
% For $k \in \mathds{N}^\ast$, 
% the \emph{initial function} is the map $i_k:  \mathbb{S}_k  \longrightarrow
% \llbracket 1, \mathsf{k} \rrbracket$ defined by $i_k\left((S^{n})_{n\in
% \mathds{N}}\right) = S^{0}$.
% \end{definition}
% Intuitivelly speaking, this function returns the first element of a strategy.
% 
% \begin{definition}%[Shift function] 
% Let $k \in \mathds{N}^\ast$. The \emph{shift
% function} is the map $\sigma_k: \mathbb{S}_k  \longrightarrow
%  \mathbb{S}_k$ defined by $\sigma_k\left((S^{n})_{n\in
% \mathds{N}}\right)=(S^{n+1})_{n\in \mathds{N}}$.
% \end{definition}
% 
% Intuitivelly speaking,
% this function shifts the strategy to the right, or, equivalently,
% removes its first element. 
% It is now possible to give some recalls in the field of the mathematical
% topology~\cite{Schwartz80} to make this document self contained.

% \subsection{Topological spaces, open sets, and neighborhoods}

% \subsection{Iteration Function, Phase Space, and Distance}

% \label{Defining}


% \begin{definition}[Function $F$]
Let
\begin{equation*}
\begin{array}{ll}
\mathcal{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.
\noindent where + and . are the Boolean ``or'' and ``and''
operators, and where $\delta(b_1,b_2)$ is 0 iff $b_1$ is equal to $b_2$.

Consider the phase space $\mathcal{X}_2$ defined as follows:
$\mathcal{X}_2 =  \mathbb{S}_N \times \mathds{B}^\mathsf{N} \times \mathbb{S}_P
\times \mathds{B}^\mathsf{P} \times \mathbb{S}_P,$
%\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{sec:ci2:notations}.

%\end{definition}

The map $\mathcal{G}_{f_0}:\mathcal{X}_2  \longrightarrow 
\mathcal{X}_2$ is defined by: $\mathcal{G}_{f_0}\left(S_p,x,S_c,m,S_m\right) = $
$$\left(\sigma_N(S_p),\mathcal{F}(i_N(S_p),x,i_P(S_c),m),\sigma_P(S_c),G_{f_0}(S_m,m),\sigma_P(S_m)\right)$$
% \begin{equation*}
% \mathcal{G}_{f_0}\left(S_p,x,S_c,m,S_m\right) =  \left(\sigma_N(S_p),\right. 
% \end{equation*}
% \begin{equation*}
% \left.\mathcal{F}(i_N(S_p),x,i_P(S_c),m),\sigma_P(S_c),G_{f_0}(m,S_m),\sigma_P(S_m)\right).
% \end{equation*}
where $i_N$ and $\sigma_N$ are given in Section~\ref{sec:common-notations}.

Then $\CID$ can be described by the
iterations of the following discrete dynamical system:
$$X^0 \in \mathcal{X}_2 \text{ and } X^{k+1}=\mathcal{G}_{f_0}(X^k).$$

%\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}%


%\subsection{Cardinality of $\mathcal{X}_2$}

% It has been proven
% in~\cite{fgb11:ip} that:

% \begin{proposition}
% %The phase space $\mathcal{X}_2$ has, at least, the cardinality of the
% %continuum.
% $\mathcal{X}_2$ 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~\cite{fgb11:ip}.
%\end{remark}


%\subsection{The Distance on $\mathcal{X}_2$}

A new distance has been defined on $\mathcal{X}_2$ as follow: 
%\begin{definition}[Distance $d_2$ on $\mathcal{X}_2$]
$\forall X,\check{X} \in \mathcal{X}_2,$ if
$X =(S_p,x,S_c,m,S_m)$ and
$\check{X} = (\check{S_p},\check{x},\check{S_c},\check{m}, \check{S_m}),$ then:
$d_2(X,\check{X}) =$
$$d_{\mathds{B}^{\mathsf{N}}}(x,\check{x})+d_{\mathds{B}^{\mathsf{P}}}(m,\check{m})
+ d_{\mathbb{S}_N}(S_p,\check{S_p})+d_{\mathbb{S}_P}(S_c,\check{S_c}) +
d_{\mathbb{S}_P}(S_m,\check{S_m}),$$
% 
% 
% \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_p,\check{S_p})+d_{\mathbb{S}_P}(S_c,\check{S_c})\\
% & + &d_{\mathbb{S}_P}(S_m,\check{S_m}),\\
% \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}
% are the two distances defined by:

$\displaystyle{d_{\mathds{B}^{\mathsf{N}}}}$ and
$\displaystyle{d_{\mathds{S}^{\mathsf{N}}}}$ are the two distances defined in
Section~\ref{sec:common-notations}.







%\subsection{Devaney's topological chaos of $\mathcal{CIS}_2$}
%\subsection{Topological security and stego-security of $\mathcal{CIS}_2$}

%To demonstrate that \CID is another example of topological chaos in the
%sense of Devaney, it has been firstly 
We have then established 
%in~\cite{fgb11:ip} 
that
$\mathcal{G}_{f_0}$ is continuous on $(\mathcal{X}_2,d_2)$. %~\cite{fgb11:ip}.
%\begin{proposition}
%$\mathcal{G}_{f_0}$ is a continuous function on $(\mathcal{X}_2,d_2)$.
%\end{proposition}

Using these materials, the security study of
\CID has been initiated in~\cite{fgb11:ip}. 

% \begin{proof}
% We use the sequential continuity.
% 
% Let $((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)_{n\in \mathds{N}}$ be a sequence of the
% phase space $\mathcal{X}_2$, which converges to $(S_p,x,S_c,m,S_m)$. We will prove
% that $\left( \mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)\right) _{n\in
% \mathds{N}}$ converges to $\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$. Let us recall
% that for all $n$, $(S_p)^n$, $(S_c)^n$ and $(S_m)^n$ are strategies, thus we
% consider a sequence of strategies (\emph{i.e.}, a sequence of sequences).
% 
% 
% As $d_2(((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n),(S_p,x,S_c,m,S_m))$ converges to 0,
% each distance
% $d_{\mathds{B}^{\mathsf{N}}}(x^n,x)$, $d_{\mathds{B}^{\mathsf{P}}}(m^n,m)$,  
% $d_{\mathbb{S}_N}((S_p)^n,S_p)$, $d_{\mathbb{S}_P}((S_c)^n,S_c)$, and
% $d_{\mathbb{S}_P}((S_m)^n,S_m)$ 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, $d_{\mathbb{S}_N}((S_p)^n,S_p)\longrightarrow 0$,
% $d_{\mathbb{S}_P}((S_c)^n,S_c)\longrightarrow 0$, and
% $d_{\mathbb{S}_P}((S_m)^n,S_m)\longrightarrow 0$, so $\exists n_4,n_5,n_6\in
%  \mathds{N},$
%  \begin{itemize}
%    \item $\forall n \geqslant n_4, d_{\mathbb{S}_N}((S_p)^n,S_p)<10^{-1}$,
%    \item $\forall n \geqslant n_5, d_{\mathbb{S}_P}((S_c)^n,S_c)<10^{-1}$,
%    \item $\forall n \geqslant n_6, d_{\mathbb{S}_P}((S_m)^n,S_m)<10^{-1}$.
%  \end{itemize}
%  
%  Let $n_7=Max(n_4,n_5,n_6)$. For
%  $n\geqslant n_7$, all the strategies $(S_p)^n$, $(S_c)^n$, and $(S_m)^n$ have the same
%  first term, which are respectively $(S_p)_0$,$(S_c)_0$ and $(S_m)_0$ :$\forall n\geqslant n_7,$
% $$((S_p)_0^n={(S_p)}_0)\wedge
% ((S_c)_0^n={(S_c)}_0)\wedge ((S_m)_0^n={(S_m)}_0).
% $$
% 
% 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 $(S_p)^n$ and $S_p$, $(S_c)^n$ and $S_c$, and $(S_m)^n$
% and $S_m$  start with the same first term.
% 
% Consequently, states of
% $\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)$ and $\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$ 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_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)\right) $ and $\left(
% \mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)\right) $ is convergent to 0.
% Let $\varepsilon >0$. 
% 
% \begin{itemize}
% \item If $\varepsilon \geqslant 1$, we have seen that distance
% between $\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)$ and 
% $\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$ 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
% $d_{\mathbb{S}_N}((S_p)^n,S_p)$, $d_{\mathbb{S}_P}((S_c)^n,S_c)$ and
% $d_{\mathbb{S}_P}((S_m)^n,S_m)$  converges to 0, we have:
% \begin{itemize}
%  \item $\exists n_9\in \mathds{N},\forall n\geqslant
%  n_9,d_{\mathbb{S}_N}((S_p)^n,S_p)<10^{-(k+2)}$,
%  \item $\exists n_{10}\in \mathds{N},\forall n\geqslant
%  n_{10},d_{\mathbb{S}_P}((S_c)^n,S_c)<10^{-(k+2)}$,
%  \item $\exists n_{11}\in \mathds{N},\forall n\geqslant
%  n_{11},d_{\mathbb{S}_P}((S_m)^n,S_m)<10^{-(k+2)}$.
% \end{itemize}
%   
% Let $n_{12}=Max(n_9,n_{10},n_{11})$
% thus after $n_{12}$, the $k+2$ first terms of $(S_p)^n$ and $S_p$, $(S_c)^n$ and
% $S_c$, and $(S_m)^n$ and $S_m$, are equal.
% \end{itemize}
% 
% 
% \noindent As a consequence, the $k+1$ first entries of the strategies of
% $\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)$ and $
% \mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$ are the same
% (due to the shift of strategies) and following the definition of
% $d_{\mathbb{S}_N}$ and $d_{\mathbb{S}_P}$:
% $$d_2\left(\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n);\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)\right)$$
% is equal to :
% $$d_{\mathbb{S}_N}((S_p)^n,S_p) + d_{\mathbb{S}_P}((S_c)^n,S_c) +
% d_{\mathbb{S}_P}((S_m)^n,S_m)$$
% 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_p)^n,x^n,(S_c)^n,m^n,(S_m)^n);\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)\right)
% \leqslant \varepsilon .
% \end{equation*}
% $\mathcal{G}_{f_0}$ is consequently continuous on $(\mathcal{X}_2,d_2)$.
% \end{proof}

% \subsection{$\mathcal{CIS}_2$ is Chaotic}

\subsection{Known Security Related Resultson \CID}


%%%% Then it has been proven% in~\cite{fgb11:ip}
%Then it has been
We have proven
that $(\mathcal{X}_2,\mathcal{G}_{f_0})$ is topologically \textbf{transitive},
\textbf{regular}, and has \textbf{sensitive dependence on initial conditions}.
Then we have the result:

\begin{theorem}%[$\mathcal{G}_{f_0}$ is a chaotic map]
\label{theo:topo-secure-cis2}
$\mathcal{G}_{f_0}$ is a chaotic map on $(\mathcal{X}_2,d_2)$ in the sense of
Devaney. Consequently the scheme \CID is topologically secure.
\end{theorem}
% 
% So the topological security is established:
% 
% \begin{theorem}%[Chaos-security of $\mathcal{CIS}_2$]
% \label{theo:cis2-chaosecurity}
% 
% \end{theorem}

Let us finally recall the following theorem about the security of \CID, which has been established too
in~\cite{fgb11:ip}.

\begin{theorem}
\label{theorem:stego-secu-cis2}
\CID is stego-secure.
\end{theorem}

% In previous works~\cite{arxive:lyapunov-CIS-2}, it has been also mathematically
% studied and evaluated the lyapunov exponent of \CID.
% Consider a dynamical system with an infinitesimal error on the initial condition
% $x_0$ . When the Lyapunov exponent is positive, this error will increase
% exponentially (situation of topological chaos), whereas it will decrease if
% $\lambda (x_0) \leq 0$.


% \begin{theorem}
% $\forall x^0 \in \mathcal{L}$, the Lyapunov exponent of \CID having
% $x^0$ for initial condition is equal to $\lambda(x^0) = \ln (NP^2)>0$.
% \end{theorem}
% %\subsection{New Security Properties}

In the following subsection, one other  security-related
property concerning the watermarking scheme \CID will be established. It is  a
qualitative property called strong transitivity.

% \subsection{Constant of Sensitivity
% Evaluation}\label{sec:quantitative-properties-ci2}
% 
% In this section, we study and evaluate one of the quantitative properties
% of the topological-security of the steganographic process \CID: the constant of
% sensitivity.
% 
% \begin{proposition}%[Constant of sensitivity of $ \mathcal{G}_{f_0}$]
% \label{prop:sensitivity-constant-evaluation}
% 
% $(\mathcal{X}_2,\mathcal{G}_{f_0})$ has sensitive dependence on initial
% conditions, and its constant of sensitivity $\delta$ is equal to $\mathsf{N-1}$.
% \end{proposition}
% 
% \begin{proof}
% Let us define $\mathcal{X}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{N}},$
% such that $$\mathcal{X}(S_p,x,S_c,m,S_m)=x$$ and
% $\mathcal{M}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{P}},$ such that
% $\mathcal{M}(S_p,x,S_c,m,S_m)=m$.
% 
% Let $\check{X}=(\check{S_p},\check{x},\check{S_c},\check{m}, \check{S_m})\in
% \mathcal{X}_2$ and $\varepsilon > 0$. We are looking for
% $\widetilde{X}=(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m},\widetilde{S_m})$
% in $\mathcal{X}_2$ such that $d_2\left(\check{X},\widetilde{X} \right) \leq
% \varepsilon$ and $\exists n_0 \in \mathds{N},
% d_2\left(\mathcal{G}_{f_0}^{n_0}(\check{X}),\mathcal{G}_{f_0}^{n_0}(\widetilde{X})
% \right) \geq N-1$.
% Let $k_{0}(\varepsilon)=\lfloor - \log _{10}(\frac{\varepsilon}{3})\rfloor +1$,
% and consider the set:
% $$\mathcal{S}_{\check{S_p},\check{S_c},\check{S_m}, k_0} =\left\{ S
% \in \mathbb{S}_N \times \mathbb{S}_P \times \mathbb{S}_P/ ((S_p)^k= \right.$$
% $$\left.(\check{S_p})^k) \wedge ((S_c)^k =(\check{S_c})^k)) \wedge
% ((S_m)^k=(\check{S_m})^k)), \forall k \leqslant k_0(\varepsilon) \right\}.$$
% Then, 
% $\forall (S_p,S_c,S_m) \in
% \mathcal{S}_{\check{S_p},\check{S_c},\check{S_m},k_0},$
% $$d_2((S_p,\check{x},S_c,\check{m},S_m);(\check{S_p},\check{x},\check{S_c},\check{m},
% \check{S_m})) < 3.\frac{\varepsilon}{3}=\varepsilon.$$
% 
% Let $\mathcal{J} =\left\{i\in
% \llbracket0,\mathsf{N-1}\rrbracket/
% \mathcal{X}\left(\mathcal{G}_{f_0}^{k_{0}}(\check{X})\right)_{i} \neq
% \mathcal{X}\left(\mathcal{G}_{f_0}^{k_{0}+N}(\check{X})\right)_{i}\right\}$ the
% set of the indexes of cells of the second component different between the two
% states $\mathcal{G}_{f_0}^{k_{0}}(\check{X})$ and
% $\mathcal{G}_{f_0}^{k_{0}+\mathsf{N}}(\check{X})$.
% 
% Let $\lambda = card(\mathcal{J})$ be the number of differences.
% 
% \begin{enumerate}
%   \item If $\lambda = \mathsf{N}$, then the point
%   $\widetilde{X}=(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m},\widetilde{S_m})$
%   defined by: \begin{enumerate}
%     \item $\widetilde{x}$ = $\check{x}$ and $\widetilde{m}$ = $\check{m}$ in
%     order to have the same second and fourth projections that $\check{X}$,
%     and thus have a possibility to be at a distance inferior to 1 of
%     $\check{X}$.
%   \item $\forall k \leqslant k_0(\varepsilon), (S_p^\ast)^k=
%   (\check{S_p})^k$, $(S_c^\ast)^k= (\check{S_c})^k$, and
%   $(S_m^\ast)^k= (\check{S_m})^k$ in order to be $\varepsilon$-close to
%   $\check{X}$.
% \item Let us now explain how to construct the three strategies $S_p^\ast$, 
% $S_c^\ast$, and $S_m^\ast$. It is done by replacing 
% $\mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_q$, $\forall q \in \llbracket0;\mathsf{N-1}\rrbracket$ with $(\lambda = \mathsf{N})$ in order to
% have a second projection $(x)$ whose cells are all different:\newline
% First of all, we must replace
% $\mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_0$ (with $q=0$):
% \begin{enumerate}
%   \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
%   \mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_q \neq
%   \check{m}_{\lambda_0}$, then we can choose $(S_p^\ast)^{k_0+1}=q$, $(S_c^\ast)^{k_0+1}=\lambda_0$,
%  $(S_m^\ast)^{k_0+1}=\lambda_0$, and so $I_q$ will be equal to $1$.
%  \item If such a $\lambda_0$ does not exist, we choose:\newline
%  $(S_p^\ast)^{k_0+1}=q$, $(S_c^\ast)^{k_0+1}=0$,
%  $(S_m^\ast)^{k_0+1}=0$,\newline $(S_p^\ast)^{k_0+2}=q$,
%  $(S_c^\ast)^{k_0+2}=0$, $(S_m^\ast)^{k_0+2}=0$ \newline and so let us notice
%  $I_q=2$.\newline
% 
% All of the $\mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_q$ are replaced similarly.
% The other terms of $S_p^\ast$,  $S_c^\ast$, and $S_m^\ast$ are constructed
% identically, and the values of $I_q$ are defined on the same way.\newline
% Let
% $\gamma = \sum_{q=0}^{\mathsf{N-1}}I_q$ with $\lambda= \mathsf{N}$. As $\forall q \in
% \llbracket0;\mathsf{N-1}\rrbracket, I_q \geq 1$ we can claim that $\gamma \geq
% \mathsf{N}$, so $\gamma >
% \mathsf{N-1}$.
% \end{enumerate}
% 
% %  \item $(S_p^\ast)^{k}=(S_p^\ast)^{j}$, $(S_c^\ast)^{k}=(S_c^\ast)^{j}$
% %  and $(S_m^\ast)^{k}=(S_m^\ast)^{j}$ where $j\leqslant k_{0}+\gamma$
% %  is satisfying $j\equiv k~[\text{mod } (k_{0}+\gamma)]$, if
% %  $k>k_{0}+\gamma$.
% 
% \item $\forall k > k_0 + \gamma, (S_p^\ast)^{k}=0$,
% $(S_c^\ast)^{k}=0$ and $(S_m^\ast)^{k}=0$ in order to completely finish defining
% $S_p^\ast$, $S_c^\ast$, and $S_m^\ast$.
% \end{enumerate}
% 
% 
% 
% Let
% $X^\ast =(S_p^\ast,\check{x},S_c^\ast,\check{m},S_m^\ast)$.\newline
% So,
% $\mathcal{G}_{f_0}^{k_{0}+\gamma}(X^\ast)=(S_p^\ast,\overline{(\check{x})},S_c^\ast,m,S_m^\ast)$\newline
% (where $\overline{(\check{x})}$ is the boolean negation of $\check{x}$), satisfies\newline
% $d_2(X^\ast;\check{X}) < \varepsilon$ (ie. $X^\ast$ is close to
% $\check{X}$), and $\forall i \in \llbracket0;\mathsf{N-1}\rrbracket,
% \mathcal{X}\left(\mathcal{G}_{f_0}^{k_{0}+\gamma}(X^\ast)\right)_{i} \neq
% \mathcal{X}\left(\mathcal{G}_{f_0}^{k_{0}+\gamma}(\check{X})\right)_{i}$, (ie.
% the iterate $k_0+\gamma$ of $X^\ast$ is away from the iterate $k_0+\gamma$ of
% $\check{X}$ with a distance greater than $\mathsf{N}$), and we have the result.
% 
% % Let $\mathcal{K} =\left\{ i \in \llbracket0;\mathsf{P-1}\rrbracket / m_i \neq
% % \mathcal{M}(X_{B})_{i}, \text{ where }\right.$\newline
% % $\left.(S_p^\ast,x_B,S_c^\ast,m,S_m^\ast)=\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_p^\ast,x_A,S_c^\ast,m_A,S_m^\ast))\right\},$\newline
% % $\mu = card(\mathcal{K})$ and $r_0 <r_1 < ... < r_{\mu-1}$ the elements of $
% % \mathcal{K}$.
%  
%  
%   \item On the contrary, if $\lambda < \mathsf{N}$, let $j_1 < ... <
%   j_{\lambda}$ be the elements of $\mathcal{J}$, and $j_0 \not\in
%   \mathcal{J}$.  Let us here build $\widetilde{X}$ and the three strategies
%   $\widetilde{S_p}$, $\widetilde{S_c}$, $\widetilde{S_m}$ as follows.
% \begin{enumerate}
%   \item $\widetilde{x}$ = $\check{x}$ and $\widetilde{m}$ = $\check{m}$ in
%     order to have the same second and fourth projections that $\check{X}$,
%     and thus have a possibility to be at a distance inferior to 1 of
%     $\check{X}$.
%   \item $\forall k \leqslant k_0(\varepsilon), (\widetilde{S_p})^k=
%   (\check{S_p})^k$, $(\widetilde{S_c})^k= (\check{S_c})^k$, and
%   $(\widetilde{S_m})^k= (\check{S_m})^k$ in order to be $\varepsilon$-close
%   to $\check{X}$.
%  \item Let us now explain how to construct the three strategies $S_p^\ast$, 
% $S_c^\ast$, and $S_m^\ast$. It is done by replacing 
% $\mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_q$, $\forall q \in \llbracket0;\mathsf{N-1}\rrbracket$ with $(\lambda = \mathsf{N})$ in order to
% have a second projection $(x)$ whose cells are all different:\newline
% First of all, we must replace
% $\mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_0$ (with $q=0$):
% \begin{enumerate}
%   \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
%   \mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_q \neq
%   \check{m}_{\lambda_0}$, then we can choose $(S_p^\ast)^{k_0+1}=q$, $(S_c^\ast)^{k_0+1}=\lambda_0$,
%  $(S_m^\ast)^{k_0+1}=\lambda_0$, and so $I_q$ will be equal to $1$.
%  \item If such a $\lambda_0$ does not exist, we choose:\newline
%  $(S_p^\ast)^{k_0+1}=q$, $(S_c^\ast)^{k_0+1}=0$,
%  $(S_m^\ast)^{k_0+1}=0$,\newline $(S_p^\ast)^{k_0+2}=q$,
%  $(S_c^\ast)^{k_0+2}=0$, $(S_m^\ast)^{k_0+2}=0$ \newline and so let us notice
%  $I_q=2$.\newline
% 
% All of the $\mathcal{X}(\mathcal{G}_{f_0}^{k_{0}}(\check{S}))_q$ are replaced similarly.
% The other terms of $S_p^\ast$,  $S_c^\ast$, and $S_m^\ast$ are constructed
% identically, and the values of $I_q$ are defined on the same way.\newline
% Let
% $\gamma = \sum_{q=0}^{\mathsf{N-1}}I_q$ with $\lambda= \mathsf{N}$. As $\forall q \in
% \llbracket0;\mathsf{N-1}\rrbracket, I_q \geq 1$ we can claim that $\gamma \geq
% \mathsf{N}$, so $\gamma >
% \mathsf{N-1}$.
% \end{enumerate}
%   \item $\widetilde{S_p}^k=(S_p^\ast)^k$, $\widetilde{S_c}^k=(S_c^\ast)^k$, and
% $\widetilde{S_m}^k=(S_m^\ast)^k$, 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 
%  $\widetilde{S_p}^{k_0+\gamma +1}=\mu_0$, $\widetilde{S_c}^{k_0+\gamma
%  +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
%  +1}=r_0$, and $J_{r_0}$ will be equal to $1$.
%  \item If such a $\mu_0$ does not exist, we choose:
% $\widetilde{S_p}^{k_0+\gamma +1}=0$, $\widetilde{S_c}^{k_0+\gamma
%  +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
%  +1}=r_0$,\newline 
% $\widetilde{S_p}^{k_0+\gamma +2}=0$, $\widetilde{S_c}^{k_0+\gamma
%  +2}=r_0$, $\widetilde{S_m}^{k_0+\gamma
%  +2}=0$,\newline 
% $\widetilde{S_p}^{k_0+\gamma +3}=0$, $\widetilde{S_c}^{k_0+\gamma
%  +3}=r_0$, $\widetilde{S_m}^{k_0+\gamma
%  +3}=0$, 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 $\widetilde{S_p}$, 
% $\widetilde{S_c}$, and $\widetilde{S_m}$ 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 $\forall k \in \mathbb{N}^\ast, \widetilde{S_p}^{k_0+ \gamma + \alpha
% + k}=(S_p)_B^{k}$, $\widetilde{S_c}^{k_0+ \gamma + \alpha + k}=(S_c)_B^{k}$, and
% $\widetilde{S_m}^{k_0+ \gamma + \alpha + k}=(S_m)_B^{k}$.
% \end{enumerate}
% \end{enumerate}
% 
% So,
% $\mathcal{G}_{f_0}^{k_{0}+\gamma+\alpha}(\widetilde{S_p},x_A,\widetilde{S_c},m_A,\widetilde{S_m})=X_B$,
% with $(\widetilde{S_p},\widetilde{S_c},\widetilde{S_m}) \in \mathcal{S}_{X_{A},
% k_0}$.
% Then $\widetilde{X} = (\widetilde{S_p},x_A,\widetilde{S_c},m_A,\widetilde{S_m})
% \in \mathcal{X}_2$ 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{Compacity of the Topological Space for \CID}

% 
% \subsection{Compacity and Strong Transitivity
% Study}\label{sec:qualitative-properties-ci2}
% 
% In this section is introduced a new topological notion called
% compacity~\cite{Schwartz80}. This concept will then be used to establish the
% strong transitivity, a new qualitative property of the topological-security of
%  watermarking process \CID.
% 
% \subsubsection{Recall About Topological Compacity}
% In order to give the topological compacity definition, it is necessary to give
% two first other preliminary definitions:
% 
% \begin{definition}
% A Hausdorff space is a topological space in which any two distinct points
% always admit disjoint neighborhoods.
% \end{definition}
% 
% 
% \begin{remark}
% All metric spaces are Hausdorff spaces.
% \end{remark}
% 
% \begin{definition}
% Let X be a topological space, then a cover $C$ of $X$ is a collection of subsets
% $(U_i)_i \in I$ of X whose union is the whole space X. In this case we say that
% C covers X, or that the sets $(U_i)_i \in I$ cover X.
% A subcover of C is a subset of C that still covers X.
% C is said to be an open cover if each of its members is an open set.
% \end{definition}
% 
% 
% \begin{definition}
% A Hausdorff topological space is called compact if each of its open covers
% has a finite subcover.
% \end{definition}
% 
% 
% 
% The sequential characterization of the compacity for
% metric spaces can now be recalled. This characterization will then be used
% in order to prove the strong transitivity of \CID:
% \begin{proposition}
% A metric space $(\mathcal{X}, d)$ is compact if and only if every infinite
% sequence  of $\mathcal{X}$ has a convergent subsequence.
% \end{proposition}
% 
% \subsubsection{Compacity of $\mathcal{X}_2$}
% 
% So, using the previous sequential characterization of  compacity, it will now be
%  proven that:
% 
% \begin{theorem}
% \label{th:compacity-X2}
% The metric space $(\mathcal{X}_2, d_2)$ is compact.
% \end{theorem}
% 
% 
% \begin{proof}
% The following proof is a constructive proof. It will be explain here how it is
% always possible to construct a convergent subsequence for any given sequence.
% 
% Let $(Y^n)_n=((S_p)^n,x^n,(S_c),m^n,(S_m)^n))_{n} \in \mathcal{X}_2^\mathds{N}$
% a sequence of points of $\mathcal{X_2}$.
% \begin{enumerate}
% \item Construction of the first term of the sequential
% limit:\newline
% \begin{enumerate}
%   \item Construction of $I_1$ and $n_x$:\newline
%   As
%   $\mathds{B}^\mathsf{N}$ is a finite set and as $\forall n \in \mathds{N}, x^n
%   \in \mathds{B}^\mathsf{N}$, $\exists \lambda_x \in \mathds{B}^\mathsf{N}$ such
%   that $\lambda_x$ appears an infinite number of times in the second projection
%   in sequence $(Y^n)_n$.  Let $n_x$ be the first index where this value
%   $\lambda_x$ appears. Let $I_1$ denote the set of terms $Y^n$ of the sequence $(Y^n)_n$ such that $x^n=x^{n_x}$. We note $I_1=\left\{ Y^n ~\big/~ n \in \mathds{N} \wedge
% x^n=x^{n_x} \right\}$.\newline
%   \item Construction of $I_2$ and $n_m$:\newline
%   As $I_1$ is an infinite
% set, $\mathds{B}^\mathsf{P}$ is a finite set, and $\forall n \in
% \mathds{N}, m^n \in \mathds{B}^\mathsf{P}$, $\exists \lambda_m \in
% \mathds{B}^\mathsf{P}$ such that $\lambda_m$ appears an infinite number of times
% in the second projection in sequence $(Y^n)_n$.  Let $n_m$ be the first index
% where this value $\lambda_m$ appears. We note $I_2=\left\{ Y^n \in I_1 ~\big/~ n
%  \in \mathds{N} \wedge x^n=x^{n_m} \wedge
% m^n=m^{n_m} \right\}$.\newline
%   \item Construction of $I_3$ and $p_0$:\newline
%   The first terms $\left((S_p)^n\right)^0$
% of strategies $(S_p)^n$ of tuples of $I_2$
% belong to $\llbracket 1, \mathsf{N} \rrbracket$, and $I_2$ is an infinite set. So, 
% $\exists k_p \in \llbracket 1, \mathsf{N} \rrbracket$
% for which there is an infinite number of strategies of $I_2$ whose first
% term is equal to $k_p$. Let $p_0$ denote the the smallest integer $n$ such that
% $Y^n \in I_2$ and $\left( (S_p)^n \right)^0 =  k_p$.
% Let $I_3 = \left\{Y^n \in I_2  ~\big/~ \right.$
% $n  \in \mathds{N} \wedge x^n=x^{p_0} \wedge m^n=m^{p_0}  \wedge 
%  \left( (S_p)^n \right)^0$
%  $\left.= \left( (S_p)^{p_0} \right)^0
% \right\}$
%   \item Construction of $I_4$ and $c_0$, $I_5$ and $m_0$:\newline
% In the same way as for the previous item, and for similar reasons, it can be
% constructed the two sets $I_4$ and $I_5= \left\{Y^n \in I_4 ~\big/~ \right.$
% $n  \in \mathds{N} \wedge x^n=x^{m_0} \wedge m^n=m^{m_0}  \wedge 
%  \left( (S_p)^n \right)^0$
%  $= \left( (S_p)^{m_0} \right)^0  \wedge  \left( (S_c)^n \right)^0=\left(
%  (S_c)^{m_0} \right)^0 $ $\left. \wedge \left(  (S_m)^n \right)^0=\left(
%  (S_m)^{m_0} \right)^0 \right\}$
% \end{enumerate}
% 
% \item The construction of all the others
% values of the sequential limit and all the other $m_k$ values used in
% the subsequence follows exactly the same pattern as
% previously.
% \end{enumerate}
% 
% 
% \noindent Let $l = \left(\left(\left((S_p)^{m_k}\right)^k,
% x^{p_0},\left((S_c)^{m_k}\right)^k,m^{p_0},\left((S_m)^{m_k}\right)^k
% \right)_{k \in \mathds{N}}\right)$,\newline then the subsequence
% $\left((S_p)^{m_k},x^{m_k},(S_c)^{m_k},m^{m_k},(S_m)^{m_k}\right)$
% converges to $l$.
% \end{proof}
% 
% \subsubsection{Strong transitivity of \CID}
% 
% Let us now give the definition of strong transitivity and recall the
% following proposition coming from works of Enrico Formenti~\cite{Formenti1998}:
% 
% \begin{definition}
% \label{def:strong-transitivity}
% A discreet dynamical system $(\mathcal{X},f)$ is \emph{strongly transitive} if
% and only if $$\forall x,y \in \mathcal{X}, \forall r>0, \exists z \in B(x,r),
% \exists n \in \mathbb{N}, f^{(n)}(z)=y.$$
% \end{definition}
% 
% 
% 
% In other words, for any pair $(x,y)$, there exists a point as
% close as we want to $x$ whose one iterate is equal to $y$.
% 
% We can recall here the following result from~\cite{Formenti1998}:
% 
% \begin{proposition}
% \label{prop:equivalence-transitivity-strong-transitivity}
% In a compact metric space, transitivity and strong transitivity are equivalent.
% \end{proposition}
% 
% So according to previous
% proposition~\ref{prop:equivalence-transitivity-strong-transitivity}, and the
% two theorems~\ref{th:compacity-X2}, and~\ref{theo:topo-secure-cis2}, we have the
% following result:
% 
% 
% 
% 
% \begin{theorem}
% \label{theo:strong-transitivity-ci2}
% $\mathcal{G}_{f_0}$ is strongly transitive on $(\mathcal{X}_2,d_2)$.
% \end{theorem}
% 
% As a consequence, \CID also strongly transitive.
% 
% 
% %\subsection{Consequences in Attack
% %Frameworks}\label{sec:qualitative-properties-ci2}
% 
% This new qualitative property of \CID allow thus \CID to better
% resists attacks in KOA, KMA,CMA and WOA setups as defined in~\cite{Cayre2008} in
% a watermarking context.
% 
% 
% 
