ajout de la partie chaos

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+\vspace{-1em}
+\begin{itemize}
+\item \cite{Wang2003}\footnote{\bibentry{Wang2003}}, \cite{Xiao20094346}\footnote{\bibentry{Xiao20094346}},\cite{Xiao20092288}\footnote{\bibentry{Xiao20092288}},\cite{Xiao20102254}\footnote{\bibentry{Xiao20102254}}.
+\item Logistic, tent,  or Arnold's cat maps: included chaotic functions of
+\alert<2>{real variables}. 
+\item Claim:  chaos properties \alert<2>{preserved} in the final  hash  function.
+\end{itemize}

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+\vspace{-1.5em}\begin{itemize}
+\item Discrete Iterative System:
+\begin{itemize}
+\item $x=(x_1,\dots,x_n)$ : $n$ components, $x_i$ in $\Bool=\{0,1\}$.
+\item A \emph{strategy}  \alert<2>{$(S^{t})^{t \in \Nats}$}: sequence of the
+  components that may be updated at time $t$.
+\item Components evolution: defined for times $t=0,1,2,\ldots$
+by:
+$$  
+\left\{
+  \begin{array}{l}   
+    \alert<2>{x^{0}}\in \Bool^{n} \textrm{ and}\\
+    x^{t+1}= (x^{t+1}_1,\dots,x^{t+1}_n) \textrm{ where } 
+    x^{t+1}_i =  
+    \left\{
+      \begin{array}{l}
+        \overline{x^{t}_i} \textrm{ if $i = S^t$}  \\
+        x^t_i \textrm{ otherwise} 
+      \end{array} 
+    \right.
+  \end{array} 
+\right.
+$$
+\end{itemize}
+\item Theoretical Results~\cite{GuyeuxThese10}\footnote{\bibentry{GuyeuxThese10}}: let $\mathcal{X}$ be 
+$ \llbracket 1 ; n \rrbracket^{\Nats} \times 
+\Bool^n$. We can define a distance $d$ on $\mathcal{X}$ and
+a function $f: \mathcal{X} \rightarrow \mathcal{X}$ from the 
+iterative process
+s.t. $f$ is a continuous and chaotic function.
+\end{itemize}
+

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+\begin{block}{Definition: Chaotic function [4]$^4$}
+Let $(\mathcal{X}; d)$ be a metric space.
+A function $f: \mathcal{X} \rightarrow \mathcal{X}$ is chaotic on $\mathcal{X}$ if:
+\begin{enumerate}
+\item $f$: topologically transitive (\textit{i.e.}, indecomposability of the system)\\
+(for any pair of open sets $U,V \subset \mathcal{X}$, $\exists k > 0 . 
+f^k (U) \cap V \neq \emptyset$)
+\\
+\onslide<2>{\alert<2>{Addressed property: preimage resistance}}.
+\item  $f$ is regular (\textit{i.e.}, fundamentally different points coexist)\\
+(the set of periodic points is dense in $\mathcal{X}$).
+\item  $f$: sensitive dependent on initial conditions (SDIC)\\
+($
+\exists \delta  > 0 .  \forall x \in  \mathcal{X}
+\textrm{ and } 
+\forall V \textrm{ neighborhood of $x$}.
+\exists y \in V \textrm{ and } 
+\exists n \ge 0 . 
+d(f^n(x); f^n(y))> \delta
+$)\\
+\onslide<2>{\alert<2>{Addressed properties: avalanche effect}}.
+\end{enumerate}
+\end{block}
+\footnote{\bibentry{devaney}}
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+ 
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