


\section{The $\mathcal{CIS}_2$ Chaotic Iteration based Steganographic Process}
\label{sec:secrypt11}



After the introduction of $\mathcal{CIW}_1$ 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{GuyeuxThese10}.
The second scheme both topologically secure and stego-secure has
been presented in the previous section.
However, this $\mathcal{CIW}_1$ process 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.
Such a study led to the definition of the $\mathcal{CIS}_2$ scheme presented here.




\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}.

\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 additionally 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 
formerly called Steganography by Chaotic Iterations and
Substitution with Mixing Message (SCISMM in short),
and 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 $\mathcal{CIS}_2$ can be described by the
iterations of the following discrete 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 document, 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 have defined in~\cite{fgb11:ip} a new distance on $\mathcal{X}_2$ as follows: 
$\forall X,\check{X} \in \mathcal{X}_2,$ 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}



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

\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}



\subsection{Correctness and completeness studies}
\label{sub:ci2:discussion}

Without attack, the $\mathcal{CIS}_2$ scheme has to ensure that the user can always extract a 
message and that this latter is the watermark, provided the user has the 
correct keys. These two demands 
correspond respectively to the study
of completeness and of correctness
for the proposed approach, which have been 
investigated in~\cite{bcfg+13:ip}. We have firstly established that,

\begin{Prop}
Let  $\Im(S_p)$ be the set (without repetitions)
  $ \{S^1_p, S^2_p, \ldots,  S^l_p\}$ of cardinality $k$, $k \leq l$.  
  This set contains all the elements of $x$ that have been modified 
  along the  $\mathcal{CIS}_2$ iteration process.
  Let us consider $\Im(S_c)_{|D}$ defined by 
  $\{S^{d_1}_c, S^{d_2}_c, \ldots,  S^{d_k}_c\}$
  where 
  $d_i$ is the last iteration that has modified the element $i \in \Im(S_p)$.   
  
Message can be extracted from the stego-content if and only if
$\Im(S_c)_{|D}=\llbracket 0 ;\mathsf{P} -1 \rrbracket$.
\end{Prop}


Under this condition, one bit of index $j$ of the original message $m^0$
is thus embedded at least twice in $x^l$. 
By counting the number of times this bit has been switched in $S_m$, the value of 
$m_j$ can be deduced in many places. 
Without attack, all these values are equal and the message is immediately 
obtained.
 After an attack,  the value of $m_j$ is obtained as mean value of all 
its occurrences. 
The scheme is thus complete.
Notice that if the cover is not attacked, the returned message is always equal to the original 
due to the definition of the mean function.


\subsection{Deciding whether a possibly attacked media is watermarked}
\label{sub:ci2:distances}

Let us consider a first media $y$ that is watermarked with a message $m$
and a second one, namely $y'$, which is an altered version of $y$, \textit{i.e.}, where 
some bits have been modified.
Let $m'$ be the message that is extracted from $y'$. 

We have checked in~\cite{bcfg+13:ip} how far the extracted message $m'$ is from $m$.
To achieve this, we have considered the set 
$M = \{ i | m_i = 1 \}$ of the Boolean vector message $m$ and similarly 
the set $M'$ for the message $m'$. 
Most of similarity measures 
depend on the functions $a$, $b$, $c$, and
$d$, all from 
$\mathbb{B}^{\mathsf{P}}  \times \mathbb{B}^{\mathsf{P}}$ to $\mathbb{N}$,
and respectively equal to
$a(m,m') = |M \cap M' |$, 
$b(m,m') = |M \setminus M' |$,
$c(m,m') = |M' \setminus M|$, and
$d(m,m') = |\overline{M} \cap \overline{M'}|$
($|S|$ and $\overline{S}$ respectively 
denote  the cardinality 
and the complementary 
of any set $S$).
In what follows $a$, $b$, $c$, and $d$ respectively stand for 
$a(m,m')$, $b(m,m')$, $c(m,m')$, and $d(m,m')$.

According 
to~\cite{rifq/others03} 
the Fermi-Dirac measure $S_{\textit{FD}}$
is the one that has the highest discrimination power, \textit{i.e.}, 
which  allows a clear separation between correlated vectors and 
uncorrelated ones. 
The measure is recalled hereafter with
respect to the previously defined scalars $a$, $b$, and $c$.

\begin{eqnarray*}
S_{\textit{FD}}(\varphi) &= &
\dfrac{F_{\textit{FD}}(\varphi) -F_{\textit{FD}}(\frac{\pi}{2})}
{F_{\textit{FD}}(0) -F_{\textit{FD}}(\frac{\pi}{2})},\\
F_{\textit{FD}}(\varphi) &= &
\dfrac{1}{1+\exp(\dfrac{\varphi - \varphi_0}{\gamma})},\\
%\varphi &= &\arctan(\dfrac{b+c}{a})
\end{eqnarray*}
\noindent where $\varphi = \arctan(\dfrac{b+c}{a})$, $\varphi_0$ is $\pi/4$, and $\gamma$ is 0.1.

The distance between $m$ and $m'$ is then computed in~\cite{bcfg+13:ip} as 
$1-S_{\textit{FD}}(m,m')$ and is thus a real number in $[0;1]$.
We have proposed in~\cite{bcfg+13:ip} that, if such a distance is lower
than a given threshold, $y'$ will be
declared as watermarked and not watermarked otherwise.
Next section presents a practical robustness evaluation of 
$\mathcal{CIS}_2$ using this decision rule. 



\subsection{Robustness study of the process}


This section is devoted to the recall of the
robustness study of the $\mathcal{CIS}_2$ scheme realized in~\cite{bcfg+13:ip}.
For the whole experiments, a set of 100 images has been randomly extracted 
from the database taken from the BOSS contest~\cite{Boss10}. 
In this set, each cover is a $512\times 512$
grayscale digital image.
The considered watermark $m$ is given in Fig.~\ref{invad}. 
Testing the robustness of the approach is achieved by successively applying
on watermarked images attacks like cropping, compression, geometric 
transformations,\ldots
Differences between $m$ and $m'$ have been
computed as described in the previous section.


We have firstly evaluate the robustness of the $\mathcal{CIS}_2$  approach by
applying different percentages of cropping, from 0.25\% to 90\%.
Results are recalled in Fig.~\ref{Fig:atck:dec}, %.
which presents effects of such an attack.
All the percentage differences are so far less than 97\% 
and thus robustness is established.



\begin{figure}[ht]
  \centering
\includegraphics[width=0.5\textwidth]{IH/iihmsp13/exp/atq-dec}\label{Fig:atq:dec:curves}%}
\caption{Cropping Results}
\label{Fig:atck:dec}
\end{figure}



Robustness against compression has then been addressed in~\cite{bcfg+13:ip},
by studying both JPEG  and JPEG 2000 image compressions.
Results are respectively presented in Fig.~\ref{Fig:atq:jpg:curves}
and Fig.~\ref{Fig:atq:jp2:curves}.
It is not hard to see that robustness is well established for 
JPEG2000 compression: for all the ratios larger than 10\%, the watermark 
is retrieved.  
However, as stated in~\cite{bcfg+13:ip}, this scheme is not robust against JPEG compression for a ratio inferior
to 90\%.
Remark that a potential solution can be to insert the
watermark in least significant coefficient of the image described in frequency
domain, for instance using either discrete cosine or with wavelet transform.
\begin{figure}[ht]
  \centering
  \subfigure[JPEG Effect]{
\includegraphics[width=0.45\textwidth]{IH/iihmsp13/exp/atq-jpg}\label{Fig:atq:jpg:curves}}
  \subfigure[JPEG 2000 Effect]{
\includegraphics[width=0.45\textwidth]{IH/iihmsp13/exp/atq-jp2}\label{Fig:atq:jp2:curves}}
\caption{Compression Results}
\label{Fig:atck:comp}
\end{figure}

Among geometric transformations, we then focused on  
rotations, \textit{i.e.}, when two opposite rotations 
of angle $\theta$ are successively applied around the center of the image.
In these geometric transformations,  angles range from 2 to 60 
degrees.  
Results are presented in Fig.~\ref{Fig:atq:rot}: thanks to an 
efficient embedding, our scheme is resistant to all
that type of attacks.



\begin{figure}[ht]
  \centering
\includegraphics[width=0.5\textwidth]{IH/iihmsp13/exp/atq-rot}\label{Fig:atq:rot:curve}%}
\caption{Rotation Attack Results}
\label{Fig:atq:rot}
\end{figure}

The first step of the $\mathcal{CIS}_2$ scheme studied in this subsection 
has defined $x$ as the LSBs of the host image, it is thus based on LSB modifications.
We have then considered in~\cite{bcfg+13:ip} two types of attacks modifying these LSB sets (see
Fig~\ref{Fig:atq:lsb}). The former consists in setting to zero a subset of this one. 
Results are expressed in Fig.~\ref{Fig:atq:lsb:curve} and 
show that the scheme is robust, unless 95\% of the LSB is erased.
In this case the image is really damaged.
The latter consists in applying again this scheme on the watermarked image 
but  with another message. 
Results of Fig.~\ref{Fig:atq:ci2:curve} show that this scheme is robust 
against that type of attack, provided the number of iterations is 
lesser than $1.75$ times the number of pixels.
With more iterations, the image is dramatically modified: more than 50\% 
of the LSB is switched.
%However, future works present ideas to tackle
%this problem.


\begin{figure}[ht]
  \centering
  \subfigure[LSB erasing effects]{
    \includegraphics[width=0.45\textwidth]{IH/iihmsp13/exp/atq-lsb}\label{Fig:atq:lsb:curve}}
  \subfigure[Applying algorithm twice]{
\includegraphics[width=0.45\textwidth]{IH/iihmsp13/exp/atq-ci2}\label{Fig:atq:ci2:curve}}

\caption{LSB Modifications}
\label{Fig:atq:lsb}
\end{figure}




\subsection{Evaluation of the embeddings}
\label{sub:roc}

A Receiver Operating Characteristic (ROC) approach has finally
 been implemented in~\cite{bcfg+13:ip}, to find
the most adapted threshold w.r.t. the separation between  
watermarked images and other ones.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{IH/iihmsp13/exp/ROC}
\end{center}
\caption{ROC Curves for DWT or DCT Embeddings}\label{fig:roc:dwt}
\end{figure}

Figure~\ref{fig:roc:dwt} recalls the obtained ROC curve. 
This latter is close to the ideal one that is without 
False Positive and False Negative answer.
The threshold with best results is  a distance equal to 0.97. 
With such a value, we can give some confidence intervals
for most of evaluated attacks. The
approach is resistant to all the cropping where percentage is less than 90\%,
to a JPEG2000 compression where quality ratio is greater than 5\%,
to all the rotation attacks,
and to LSB erasing when less than 95\% LSBs are set to 0.



\subsection{Lyapunov evaluation of $\mathcal{CIS}_2$}
\label{sec:lyapCIS2}

We finally close the study of the  $\mathcal{CIS}_2$ process by 
recalling the way we evaluated its Lyapunov exponent in Secrypt13~\cite{bfg13:ip}.


\subsubsection{A topological semi-conjugacy between $\mathcal{X}_2$ and $\mathds{R}$}
\label{sec:topological-semiconjugacy}

In this section, by using a topological
semi-conjugacy, we recall that $\mathcal{CIS}_2$ modeled by $\mathcal{G}_{f_0}$ on
$\mathcal{X}$ can be described as iterations on a real interval. 
To do so, new notations and terminologies must be introduced.

Let $\mathcal{X}_{(\mathsf{N};\mathsf{P})} =
\mathbb{S}_N \times \mathds{B}^\mathsf{N} \times \mathbb{S}_P \times \mathds{B}^\mathsf{P} \times \mathbb{S}_P$.
In what follows and for easy understanding, 
we will assume that $\mathsf{N}=3$ and $\mathsf{P}=2$.  
So $\mathsf{N}+\mathsf{P} =
5$  and $\mathsf{N}\mathsf{P}^2 = 12$. However, an equivalent formulation of the
following can be easily obtained by replacing the bases $5$ and $12$ by any base
$(\mathsf{N}+\mathsf{P})$ and $(\mathsf{N}\mathsf{P}^2)$.
 $\mathsf{N}$ has only
to be greater than $\mathsf{P}$. 

\begin{Def}
\label{def:psi-function}
The function $\psi: \llbracket 1,\mathsf{N}\rrbracket \times \llbracket
1,\mathsf{P}\rrbracket  \times \llbracket
1,\mathsf{P}\rrbracket \rightarrow \llbracket
0,\mathsf{N}\mathsf{P}^2-1\rrbracket$ is defined by:
$\psi \left(S_p^i,S_c^i,S_m^i\right)=(S_p^i-1) \mathsf{P}^2 + (S_c^i-1)
\mathsf{P} + (S_m^i-1)$.
\end{Def}
This function aims to convert a strategy of triplets
in a simple strategy of integers expressed in a different basis, see Table~\ref{tablePsi}. 
Obviously, $\psi$ is a bijective
function, the reverse operation will be denoted by $\psi^{-1}$. The three
projections of $\psi^{-1}$ are denoted by: $\psi^{-1}_1\left(\psi \left(S_p^i,S_c^i,S_m^i\right)\right)=S_p^i$, $\psi^{-1}_2\left(\psi \left(S_p^i,S_c^i,S_m^i\right)\right)=S_c^i$, and $\psi^{-1}_3\left(\psi \left(S_p^i,S_c^i,S_m^i\right)\right)=S_m^i$.


\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Base&Base&Base&Base\\
$\mathsf{N}=3$ & $\mathsf{P}=2$ &
$\mathsf{P}=2$&$\mathsf{N}\mathsf{P}^2=12$\\
\hline
\hline
$S_p^i$ & $S_c^i$ & $S_m^i$&$\psi \left(S_p^i,S_c^i,S_m^i\right)$\\
\hline
\hline
1 & 1 & 1&0\\
1 & 1 & 2&1\\
1 & 2 & 1&2\\
1 & 2 & 2&3\\
2 & 1 & 1&4\\
2 & 1 & 2&5\\
2 & 2 & 1&6\\
2 & 2 & 2&7\\
3 & 1 & 1&8\\
3 & 1 & 2&9\\
3 & 2 & 1&10\\
3 & 2 & 2&11\\
\hline
\end{tabular}
\end{center}
\caption{Some values for $\psi$~(see Definition~\ref{def:psi-function}).}
\label{tablePsi}
\end{table}


\begin{Def}
Let us define
$\varphi: \mathcal{X}_{(\mathsf{3};\mathsf{2})} = \mathbb{S}_{3} \times
\mathds{B}^\mathsf{3} \times \mathbb{S}_2 \times \mathds{B}^\mathsf{2} \times
\mathbb{S}_2  \longrightarrow  \big[ 0, 2^{5} \big[$,
as follows. If $\left(S_p,E,S_c,M,S_m\right) = \left((S_p^0, S_p^1, \hdots );
(E_0,E_1,E_2, E_3);(S_c^0, S_c^1, \hdots );(M_0,M_1);(S_m^0, S_m^1, \hdots
)\right),$ then
 $\varphi \left(S_p,E,S_c,M,S_m\right)$ is the real number:
 \begin{itemize}
 \item whose integral part $e$ is $\displaystyle{\sum_{k=0}^2 2^{4-k}
 E_k + \sum_{k=3}^4 2^{4-k}M_{k-3}}$, that
 is, the binary digits of $e$ are $E_0 ~ E_1 ~ E_2 ~ M_0 ~ M_1$.
 \item whose decimal part $s$ is equal to:
   $s = 0,\psi
 \left(S_p^0,S_c^0,S_m^0\right)~ \psi
 \left(S_p^1,S_c^1,S_m^1\right)~ \psi
 \left(S_p^2,S_c^2,S_m^2\right)~ \hdots$
 $ = \sum_{k=1}^{+\infty} 12^{-k}
 S^{k-1}.$ $s$ is thus expressed in base $12$.
 \end{itemize}
\end{Def}



As notified in~\cite{bfg13:ip}, $\varphi$ realizes the association between a point of
$\mathcal{X}_{(\mathsf{3};\mathsf{2})}$ and a real number into $\big[ 0, 2^{5}
\big[$. We must now translate the steganographic process
$\mathcal{CIS}_2$, which is represented by $\mathcal{G}_{f_0}$, as	 %based  on chaotic
iterations on this real interval. To do so, 
two intermediate functions over $\big[ 0, 2^{5}
\big[$ denoted by $e$ and $s$ has been introduced in~\cite{bfg13:ip}.

\begin{Def}
\label{def:e et s}
 Let $x \in \big[ 0, 2^{5} \big[$ and:
 \begin{itemize}
 \item $e_0, \hdots, e_4$ the binary digits of the integral part of $x$:
 $\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{4} 2^{4-k} e_k}$.
 \item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, expressed in base $12$, where
 the chosen decimal
 decomposition of $x$ is the one that does not have an infinite number of
 $11$: 
 $\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k
 12^{-k-1}}$.
 \end{itemize}
$e$ and $s$ are thus defined as follows:
\begin{equation*}
\begin{array}{cccl}
e: & \big[ 0, 2^{5} \big[ & \longrightarrow & \mathds{B}^{3} \times
\mathds{B}^{2}\\ & x & \longmapsto & ((e_0,e_1,e_2);(e_3,e_4))
\end{array}
\end{equation*}
and
\begin{equation*}
 \begin{array}{cccc}
s: & \big[ 0, 2^{5} \big[ & \longrightarrow & \llbracket 0, 11
\rrbracket^{\mathds{N}} \\
 & x & \longmapsto & (s^k)_{k \in \mathds{N}}\\
\end{array}
\end{equation*}
\end{Def}

We have thus been able to define the function $g$, whose goal is to translate
the steganographic process $\mathcal{CIS}_2$ represented by $\mathcal{G}_{f_0}$
on an interval of $\mathds{R}$~\cite{bfg13:ip}.

\begin{Def}\label{def:function-g-on-R}
$g:\big[ 0, 2^{5} \big[ \longrightarrow \big[ 0, 2^{5} \big[$ is
such that $g(x)$ is the real number of $\big[ 0, 2^{5} \big[$
defined below:
\begin{itemize}
 \item its integral part has a binary decomposition equal to $e_0',
 \hdots,
 e_4'$, with $\forall i \in \llbracket 0,2 \rrbracket$:
  \begin{equation*}
 e_i' = \left\{
 \begin{array}{ll}
 e(x)_i & \textrm{ if } i \neq \psi^{-1}_1\left(s^0\right)\\
 e(x)_{2+\psi^{-1}_2\left(s^0\right)}  & \textrm{ if } i =
 \psi^{-1}_1\left(s^0\right)\\
 \end{array}
 \right.
 \end{equation*}
 and $\forall i \in \llbracket 3,4 \rrbracket$:
  \begin{equation*}
 e_i' = \left\{
 \begin{array}{ll}
 e(x)_i & \textrm{ if } i \neq \psi^{-1}_3\left(s^0\right)\\
 e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = \psi^{-1}_3\left(s^0\right),\\
 \end{array}
 \right.
 \end{equation*}
\item
whose decimal part is $s(x)^1, s(x)^2, \hdots$
\end{itemize}
\end{Def}



In other words, if $x = \displaystyle{\sum_{k=0}^{4} 2^{4-k} e_k + 
\sum_{k=0}^{+\infty} s^{k} ~12^{-k-1}}$, then:
$$g(x) = 
\displaystyle{\sum_{k=0}^{2} 2^{4-k} \left[e_k
\left(\delta(k,\psi^{-1}_1(s^0))+1 \textrm{ (mod 2)}\right)+e_{2+\psi^{-1}_2(s^0)}\left(\delta(k,\psi^{-1}_1(s^0))\right)\right]}
$$
$$
 + \displaystyle{\sum_{k=3}^{4} 2^{4-k} (e_k +
 \delta(k,\psi^{-1}_3(s^0) \textrm{ (mod 2)})+\sum_{k=0}^{+\infty} s^{k+1}
 12^{-k-1}}, $$
where $\delta$ is the discrete Boolean metric
introduced previously.


Numerous metrics can be defined on the set $\big[ 0, 2^{5} \big[$, the
most
usual one being the Euclidian distance
$\Delta(x,y) = |y-x|^2$.
However, this Euclidian distance does not reproduce exactly the notion of
proximity induced by distance $d_2$ on $\mathcal{X}_2$ introduced in a previous
section, which is more relevant for the targetted applications. 
Indeed $d_2$ is richer than $\Delta$, this is why we have introduced the 
following map in~\cite{bfg13:ip}.

\begin{Def}
Given $x,y \in \big[ 0, 2^{5} \big[$,
$D$ denotes the function from $\big[ 0, 2^{5} \big[^2$ to $\mathds{R}^
+$
defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s
\left(s(x),s(y)\right)$,
where:
\begin{center}
$\displaystyle{D_e(e,\check{e}) = \sum_{k=0}^\mathsf{4} \delta (e_k,
\check{e}_k)}$, ~~and~ $\displaystyle{D_s(s,\check{s}) = \sum_{k = 1}^
\infty
\dfrac{|s^k-\check{s}^k|}{12^k}}$.
\end{center}
\end{Def}

We have thus proven in~\cite{bfg13:ip} that,

\begin{Prop}
$D$ is a distance on $\big[ 0, 2^{5} \big[$.
\end{Prop}



The convergence of sequences according to $D$ is not the same than the
usual
convergence related to the Euclidian metric. For instance, if $x^n \to x
$
according to $D$, then necessarily the integral part of each $x^n$ is
equal to
the integral part of $x$ (at least after a given threshold), and the
decimal
part of $x^n$ corresponds to the one of $x$ ``as far as required''.
$D$ is richer and more refined than the Euclidian distance, and thus is
more precise.


$\varphi$ has been constructed in order to be continuous and onto,
so
we obtained the following theorem in~\cite{bfg13:ip}.

\begin{Th}
The steganographic process $\mathcal{CIS}_2$ represented by
$\left(\mathcal{G}_{f_0},\mathcal{X}_2\right)$ can be considered as simple iterations on
$\mathds{R}$, which is illustrated by the semi-conjugacy given
below:
\begin{equation*}
\begin{CD}
\left(~\mathcal{X}_{(3;2)}, d_2~\right) @>\mathcal{G}_{f_0}>>
\left(~\mathcal{X}_{(3;2)}, d_2~\right)\\
    @V{\varphi}VV                    @VV{\varphi}V\\
\left( ~\big[ 0, 2^{5} \big[, D~\right)  @>>g> \left(~\big[ 0, 2^{5}
\big[,
D~\right)
\end{CD}
\end{equation*}
\end{Th}


In other words, $\mathcal{X}_2$ is approximately equal to $\big[ 0, 2^{
\mathsf{N}+\mathsf{P}}
\big[$.
We have thus remarked in~\cite{bfg13:ip} that,

\begin{Prop}
\label{Prop:derivabilite-de-g}
The process $\mathcal{CIS}_2$ represented by $g$ defined on $\mathds{R}$ has
derivatives of all orders on
$\big[ 0, 2^{5} \big[$, except on the 385 points in $I$ defined by:
$I=\left\{
\dfrac{n}{12} ~\big/~ n \in \llbracket 0;2^{5}\times 12\rrbracket
\right\}.$

Furthermore, on each interval of the form $\left[ \dfrac{n}{12},
\dfrac{n+1}{12}\right[$, with $n \in \llbracket 0;2^{5}\times 12
\llbracket$,
$g$ is a linear function having a slope equal to 12: $\forall x \notin
I,
g'(x)=12$.
\end{Prop}

We are now able to recall the way to evaluate the Lyapunov exponent of $\mathcal{CIS}_2$.

\subsubsection{Topological security of $\mathcal{CIS}_2$ on $\mathds{R}$}
\label{section:topo-secu-cis2-R}

$\mathcal{CIS}_2$ represented by the function $\mathcal{G}_{f_0}$ on
$\mathcal{X}_2$ is topologically secure, that is to say $\left(\mathcal{G}_{f_0}, \mathcal{X}_2\right)$ is chaotic in the sense of Devaney. We can deduce
 the same property for $\mathcal{CIS}_2$ represented by the $g$ function on 
$\mathds{R}$ for the order topology. Indeed
$\left(\mathcal{G}_{f_0}, \mathcal{X}_2\right)$ and $\left(g, [ 0,
2^{5} [_D\right)$ are semi-conjugate by $\varphi$ as recalled below. 
So $\left(g, [ 0, 2^{5} [_D\right)$ is a chaotic system
according to Devaney, because the semi-conjugacy preserves this character~\cite{Formenti1998}. 
However the topology generated by $D$ is finer than the topology
generated by the Euclidean distance $\Delta$, which is the order topology.
This is why we have proven in~\cite{bfg13:ip} that,
\begin{Th}
\label{theo:chaos-and-fineness}
Let $\mathcal{X}$ be a set, and $\tau, \tau'$ two topologies on
$\mathcal{X}$ such that $\tau'$ is finer than $\tau$. Let $f:\mathcal{X}
\to \mathcal{X}$, continue for both $\tau$ and $\tau'$.

If $(\mathcal{X}_{\tau'},f)$ is chaotic in the sense of Devaney, then
$(\mathcal{X}_\tau,f)$ is also chaotic.
\end{Th}

Finally, according to Theorem~\ref{theo:chaos-and-fineness}, we have
deduced in~\cite{bfg13:ip} that
the steganographic process $\mathcal{CIS}_2$ represented by $g$ is 
chaotic in the sense of Devaney, for the order topology on $\mathds{R}$.
Having these assertions in mind, we have then formulated the following theorem:
\begin{Th}
\label{th:cis2-topo-order}
The steganographic process $\mathcal{CIS}_2$ represented by $g$ on $\mathds{R}$
is chaotic in the sense of Devaney, when the usual topology of $\mathds{R}$ is
used (the order topology).
\end{Th}


This result is weaker than Theorem~\ref{theo:chaotic}, which
establishes the chaotic property of $\mathcal{CIS}_2$ for a finer topology. It is 
as if the chaos observed using usual tools like the Euclidian distance
is still preserved when considering more powerful tools (higher resolution, 
\emph{i.e.}, finer topologies).
The result contained in Theorem~\ref{th:cis2-topo-order} is however interesting,
as it confirms that approach followed in~\cite{bfg13:ip} does not lead to deflated properties.

Indeed, our studies take place in a system other than the one usually considered in 
computer science ($\mathcal{X}_2$ instead of $\mathds{R}$), 
in order to be as close as possible to the targetted computer machines.
By doing so, we prevent from any loss of chaotic properties 
when computing the scheme written in mathematical terms.
However, it might be feared that the choice of a
discrete mathematics approach leads to a disorder of lower quality. 
In other words, perhaps we have avoided a
situation of great disorder \emph{lost} during the computation into finite machines.
But the cost of such success may be to obtain a weaker disorder ?
Theorem~\ref{th:cis2-topo-order} proves  exactly the contrary.



\subsubsection{Evaluation of the Lyapunov exponent}
\label{section:lyapunov-exponent-evaluation}

Let $\mathcal{L} = \left\{ x^0 \in \big[ 0, 2^{5} \big[ ~ \big/ ~ \forall n
\in \mathds{N}, x^n \notin I \right\}$, where $I$ is the set of points in the
real interval where $g$ is not differentiable (as it is explained in Proposition
\ref{Prop:derivabilite-de-g}). Then~\cite{bfg13:ip}.

\begin{Th}
$\forall x^0 \in \mathcal{L}$, the Lyapunov exponent of $\mathcal{CIS}_2$ having
$x^0$ for initial condition is equal to $\lambda(x^0) = \ln (12)>0$.
\end{Th}


\begin{Rem}
The set of initial conditions for which this exponent is not calculable is countable.
This is indeed the initial conditions such that an iteration value will be a
number having the form $\dfrac{n}{12}$, with $n\in \mathds{N}$. 
Moreover, for a system having $\mathsf{N}+\mathsf{P}$
cells (a number of LSCs equal to $\mathsf{N}$ and a secret message to embed of
width equal to $\mathsf{P}$), we will find, \emph{mutatis mutandis}, an infinite
uncountable set of initial conditions $x^0 \in \left[0;2^\mathsf{N+P}\right[$
such that $\lambda (x^0) = \ln (\mathsf{N}\mathsf{P}^2)$.
\end{Rem}



So, it is possible to make the Lyapunov exponent of the
scheme $\mathcal{CIS}_2$ as large as possible, depending on the number of least
significant coefficients of the cover media we decide to consider, and 
on the width of the message to embed. As proven in~\cite{gfb10:ip}, a large Lyapunov exponent makes
it impossible to achieve the well-known ``Estimated Original Attacks''~\cite{Cayre2008}.
