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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}
--- /dev/null
+\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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