


\section{Chaotic Iterations for Steganography: Stego-security and chaos-security~\cite{fgb11:ip}}
\label{sec:secrypt11}
The next investigations in steganography since our thesis was published in
Secrypt 11~\cite{fgb11:ip}. The contribution is summarized thereafter.

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


After the study published in~\cite{gfb10:ip}, there were only two information hiding 
schemes being both stego-secure and topologically secure.
The first one is based on a spread spectrum technique called Natural Watermarking.
It is stego-secure when its parameter $\eta$ is equal to $1$ \cite{Cayre2008}. 
Unfortunately, this scheme is neither robust, nor able to face an attacker 
in KOA and KMA setups, due to its lack of expansiveness~\cite{gfb10:ip}.
The second scheme both topologically secure and stego-secure, presented in the previous section, is based on chaotic iterations.
However, it allows to embed securely only one bit per embedding parameters. 
The objective of~\cite{fgb11:ip} was to improve the scheme studied in~\cite{gfb10:ip}, in such a way that more than one bit can be embedded.





\subsection{The improved algorithm}
\label{section:Algorithm}

\label{section:IH based on CIs}
\label{sec:topological}

Let us firstly recall the notations and terminologies introduced 
in~\cite{fgb11:ip}, which extend the ones presented in Chapter~\ref{chpt:recalls}.

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



Intuitively, a strategy-adapter aims at generating a strategy 
$(S^t)^{t \in \mathds{N}}$ where each term $S^t$ belongs to 
$\llbracket 1, n \rrbracket$. 



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


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

Let us additionnaly recall the following notations.
\begin{itemize}
  \item $x^0 \in \mathbb{B}^\mathsf{N}$ the $\mathsf{N}$ least
  significant coefficients of a given cover media $C$.% $X$ be the initial state $X_0$,
  \item $m^0 \in \mathbb{B}^\mathsf{P}$ is the watermark to embed into $x^0$.
  \item $S_1 \in \mathbb{S}_N$ is a strategy called \textbf{place strategy}, giving the location (LCS) where to insert the message at each iteration.
  \item $S_2 \in \mathbb{S}_P$ is a strategy called \textbf{choice strategy}, providing which bits from the message must be inserted at the given iteration.
  \item Lastly, $S_3 \in \mathbb{S}_P$ is a strategy called \textbf{mixing strategy}, as it is required for chaos to mix the message at each
  iteration.
\end{itemize}




The information hiding scheme published in~\cite{fgb11:ip} was 
called Steganography by Chaotic Iterations and
Substitution with Mixing Message (SCISMM in short,
which has been renamed $\mathcal{CIS}_2$ in later
publications). It is defined by
$\forall (n,i,j) \in
\mathds{N}^{\ast} \times \llbracket 0;\mathsf{N-1}\rrbracket \times \llbracket
0;\mathsf{P-1}\rrbracket$:

\begin{equation*}
\left\{
\begin{array}{l}
x_i^n=\left\{
\begin{array}{ll}
x_i^{n-1} & \text{ if }S_1^n\neq i \\
m_{S_2^n} & \text{ if }S_1^n=i.
\end{array}
\right.
\\
\\
m_j^n=\left\{
\begin{array}{ll}
m_j^{n-1} & \text{ if }S_3^n\neq j \\
\overline{m_j^{n-1}} & \text{ if }S_3^n=j.
\end{array}
\right.
\end{array}
\right.
\end{equation*}



The stego-content is the Boolean vector $y = x^P \in
\mathbb{B}^\mathsf{N}$. 
  
  \subsection{Security study of the $\mathcal{CIS}_2$}

After having introduced the $\mathcal{CIS}_2$, we have studied its security in~\cite{fgb11:ip}.


\subsubsection{Stego-security}

We have proven in~\cite{fgb11:ip} that,

\begin{Prop}
$\mathcal{CIS}_2$ is stego-secure.
\end{Prop}

\begin{Pre}
See~\cite{fgb11:ip}.
\end{Pre}



\subsubsection{Topological security}


\paragraph{Topological model}

We have firstly proven in~\cite{fgb11:ip} that $\mathcal{CIS}_2$ can be modeled as a
dynamical system in a topological space, as follows.
Let
\begin{equation*}
\begin{array}{ll}
F: & \llbracket0;\mathsf{N-1}\rrbracket \times
\mathds{B}^{\mathsf{N}}\times \llbracket0;\mathsf{P-1}\rrbracket  \times
\mathds{B}^{\mathsf{P}}
\longrightarrow \mathds{B}^{\mathsf{N}} \\
 & (k,x,\lambda,m)  \longmapsto  \left( \delta
 (k,j).x_j+\overline{\delta (k,j)}.m_{\lambda}\right) _{j\in
 \llbracket0;\mathsf{N-1}\rrbracket}\\
 \end{array}%
\end{equation*}%
%\end{Def}

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

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

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


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

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

\noindent Then $\mathcal{CIS}_2$ can be described by the
iterations of the following discret dynamical system:


\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*}%

Then, by comparing $\mathcal{X}_2$ and the phase space $\mathcal{X}$ formerly introduced in 
this manuscript, we have verified in~\cite{fgb11:ip} that.

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


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


\paragraph{A new distance on $\mathcal{X}_2$}

We define a new distance on $\mathcal{X}_2$ as follow: 
$\forall X,\check{X} \in \mathcal{X}_2,$ if $X =(S_1,x,S_2,m,S_3)$ and $\check{X} =
(\check{S_1},\check{x},\check{S_2},\check{m}, \check{S_3}),$ then:
\begin{equation*}
\begin{array}{lll}
d_2(X,\check{X}) & = &
d_{\mathds{B}^{\mathsf{N}}}(x,\check{x})+d_{\mathds{B}^{\mathsf{P}}}(m,\check{m})\\
& + &
d_{\mathbb{S}_N}(S_1,\check{S_1})+d_{\mathbb{S}_P}(S_2,\check{S_2})+d_{\mathbb{S}_P}(S_3,\check{S_3}).
\end{array}
\end{equation*}


\paragraph{Continuity of $\mathcal{CIS}_2$}

To prove that $\mathcal{CIS}_2$ is another example of topological chaos in the
sense of Devaney, $\mathcal{G}_{f_0}$ must be continuous on the metric
space $(\mathcal{X}_2,d_2)$. We thus have proven in~\cite{fgb11:ip} that,

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


\paragraph{$\mathcal{CIS}_2$ is chaotic}
\label{section:chaos-security}



In conclusion, $(\mathcal{X}_2,\mathcal{G}_{f_0})$ has been proven to be topologically transitive, regular,
and sensitive dependence on initial conditions. Then we have the result~\cite{fgb11:ip} that.

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

So we can claim that $\mathcal{CIS}_2$ is topologically secure.
\end{Th}
