[hdrcouchot.git] / oxford.tex

\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.