



In this section is explained how CIs can be used as an information hiding scheme.

\subsection{Most and Least Significant Coefficients}\label{sec:msc-lsc}



We first take into account the fact that, in a watermarking process, terms of the original content $x$ that may be replaced by terms issued
from the watermark $y$ 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,
depending on its position $t$ \cite{todo}.

\begin{definition}[Signification function]
A \emph{signification function} is a real sequence 
$(u^k)^{k \in \mathds{N}}$. % with a limit equal to 0.
\end{definition}


\begin{example}\label{Exemple LSC}
Let us consider a set of    
grayscale images stored into portable graymap format (P3-PGM):
each pixel ranges between 256 gray levels, \textit{i.e.},
is memorized with eight bits.
In that context, we consider 
$u^k = 8 - (k  \mod  8)$  to be the $k$-th term of a signification function 
$(u^k)^{k \in \mathds{N}}$. 
Intuitively, in each group of eight bits (\textit{i.e.}, for each pixel) 
the first bit has an importance equal to 8, whereas the last bit has an
importance equal to 1. This is compliant with the idea that
changing the first bit affects more the image than changing the last one.
\end{example}

\begin{definition}[Significance of coefficients]
\label{def:msc,lsc}
Let $(u^k)^{k \in \mathds{N}}$ be a signification function, 
$m$ and $M$ be two reals s.t. $m < M$. 
\begin{itemize}
\item The \emph{most significant coefficients (MSCs)} of $x$ is the finite 
  vector  $$u_M = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k 
    \geqslant M \textrm{ and }  k \le \mid x \mid \right);$$
 \item The \emph{least significant coefficients (LSCs)} of $x$ is the 
finite vector 
$$u_m = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k 
  \le m \textrm{ and }  k \le \mid x \mid \right);$$
 \item The \emph{passive coefficients} of $x$ is the finite vector 
   $$u_p = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } 
u^k \in ]m;M[ \textrm{ and }  k \le \mid x \mid \right).$$
 \end{itemize}
 \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.
These two definitions are illustrated on Figure~\ref{fig:MSCLSC}, where the significance function $(u^k)$ is defined as in Example \ref{Exemple LSC}, $M=5$, and $m=6$.

\begin{figure*}[htb]
\begin{center}

\begin{minipage}[b]{.98\linewidth}
  \centering
  \centerline{\includegraphics[width=6.cm]{img/lena512}}
   %\centerline{\epsfig{figure=img/lena512.pdf,width=4cm}}
  \centerline{(a) Original Lena.}
\end{minipage}
\begin{minipage}[b]{.49\linewidth}
  \centering
    \centerline{\includegraphics[width=6.cm]{img/lena_msb_678}}
   %\centerline{\epsfig{figure=img/lena_msb_678.pdf,width=4cm}}
  \centerline{(b) MSCs of Lena.}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\linewidth}
  \centering
    \centerline{\includegraphics[width=6.cm]{img/lena_lsb_1234_facteur17}}
  
 %\centerline{\epsfig{figure=img/lena_lsb_1234_facteur17.pdf,width=4cm}}
  \centerline{(c) LSCs of Lena ($\times 17$).}
\end{minipage}
%
\caption{Most and least significant coefficients of Lena.}
\label{fig:MSCLSC}
\end{center}

\end{figure*}










\subsection{Presentation of the Scheme}

We have proposed in \cite{guyeux10ter} to use chaotic iterations as an information hiding scheme, as follows. 
Let:
\begin{itemize}
  \item $(K,N) \in [0;1]\times \mathds{N}$ be an embedding key,
  \item $X \in \mathbb{B}^\mathsf{N}$ be the $\mathsf{N}$ LSCs of a cover $C$,% $X$ be the initial state $X_0$,
  \item $(S^n)_{n \in \mathds{N}} \in \llbracket 0, \mathsf{N-1}
  \rrbracket^{\mathds{N}}$ be a strategy, which depends on the message to hide $M \in [0;1]$ and $K$,
  \item $f_0 : \mathbb{B}^\mathsf{N} \rightarrow \mathbb{B}^\mathsf{N}$ be the vectorial logical negation.
\end{itemize}


So the watermarked media is $C$ whose LSCs are replaced by $Y_K=X^{N}$, where:

\begin{equation*}
\left\{
 \begin{array}{l}
X^0 = X\\
\forall n < N, X^{n+1} = G_{f_0}\left(X^n\right).\\
\end{array} \right.
\end{equation*}


Two ways to generate $(S^n)_{n \in \mathds{N}}$ are given in \cite{trx}, namely
Chaotic Iterations with Independent Strategy~(CIIS) and Chaotic Iterations with Dependent
Strategy~(CIDS). 
In CIIS, the strategy is independent from the cover media $C$, whereas in CIDS the strategy will be dependent on $C$. 



Revoir ce qui suit.

As we will use the CIIS strategy in this document, we recall it below.
Finally, MSCs are not used here, as we do not consider the case of authenticated watermarking.

\subsection{CIIS Strategy}

Let us firstly give the definition of the Piecewise Linear Chaotic Map~(PLCM, see~\cite{Shujun1}):

%\begin{definition}%[PLCM]
%The \emph{Piecewise Linear Chaotic Map} is defined by
\begin{equation*}
F(x,p)=\left\{
 \begin{array}{ccc}
x/p & \text{if} & x \in [0;p], \\
(x-p)/(\frac{1}{2} - p) & \text{if} & x \in \left[ p; \frac{1}{2} \right],
\\
F(1-x,p) & \text{else,} & \\
\end{array} \right.
\end{equation*}
%\end{definition}


\noindent where $p \in \left] 0; \frac{1}{2} \right[$ is a ``control parameter''.

The general term of the strategy $(S^n)_n$ in CIIS setup is defined by
the following expression: $S^n = \left \lfloor \mathsf{N} \times K^n \right \rfloor +
1$, where:

\begin{equation*}
\left\{
 \begin{array}{l}
p \in \left[ 0 ; \frac{1}{2} \right] \\
K^0 = M \otimes K\\
K^{n+1} = F(K^n,p), \forall n \leq N_0, \end{array} \right.
\end{equation*}



\noindent in which $\otimes$ denotes the bitwise exclusive or (XOR) between two floating part numbers (\emph{i.e.}, between their binary digits representations). 


\subsection{CIDS Strategy}

The same notations as above are used.
We define CIDS strategy as follows: $\forall k \leqslant N$,  
\begin{itemize}
\item if $k \leqslant \mathsf{N}$ and $X^k = 1$, then $S^k=k$,
\item else $S^k=1$.
\end{itemize}
In this situation, if $N \geqslant \mathsf{N}$, then only two watermarked contents are possible with the information hiding scheme proposed previously, namely: $Y_K=(0,0,\cdots,0)$ and $Y_K=(1,0,\cdots,0)$.


\subsection{PSNR evaluation}\label{section:psnr-ci-1}

To realize the evaluation of the PSNR of $CI_1$, we have used the same
architecture as described in
Section~\ref{section:architecture-presentation}~\vpageref{section:architecture-presentation}.

It has to be noted that in the $CI_1$ watermarking process, the embedding key
correspond to the strategy associated to the number of iterations.  

Executing the $CI_1$ process on 1000 images, on the one hand with a  constant
embedding key and one the other hand with a key generated randomly we have
obtain the results described in the
table~\ref{table:psnr-ci1}~\vpageref{table:psnr-ci1}, an average of and a standard deviation of  has been obtained for the PSNR.\newline



\begin{table*}
\begin{center}

 \begin{tabular}{|c||c|c|c|}
  \hline
   \textbf{Strategy} &  \textbf{PSNR average} &  \textbf{PSNR standard
   deviation}\\
    \hline 
    \hline 
 \textbf{Constant} &  $21.6653$ &  $0.03604$  \\
  \hline
 \textbf{Generated randomly}  & $13.3909$ & $10.5690$  \\
  \hline
 
  \hline 
  \end{tabular}
  \caption{Evaluation of the PSNR of the watermarking process $CI_1$.(Tests
  realized on 1000 images)}
  \end{center}
  \label{table:psnr-ci1}
  \end{table*}

Let us now focus on some security aspects of information hiding schemes. In particular, we aim to explain why chaos is relevant in this field.
