Interet des itérations chaotiques pour générer des nombre alea\\
Interet de générer des nombres alea sur GPU
+\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
...
-% >>>>>>>>>>>>>>>>>>>>>> Basic recalls <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
+
\section{Basic Recalls}
\label{section:BASIC RECALLS}
This section is devoted to basic definitions and terminologies in the fields of topological chaos and chaotic iterations.
Finally, it has been established in \cite{guyeux10} that,
\begin{proposition}
-$G_{f}$ must be continuous in the metric
-space $(\mathcal{X},d)$.
+Let $f$ be a map from $\mathds{B}^n$ to itself. Then $G_{f}$ is continuous in the metric space $(\mathcal{X},d)$.
\end{proposition}
-The chaotic property of $G_f$ has been firstly established for the vectorial Boolean negation \cite{guyeux10}. To obtain a characterization, we have introduced the notion of asynchronous iteration graph recalled bellow.
+The chaotic property of $G_f$ has been firstly established for the vectorial Boolean negation \cite{guyeux10}. To obtain a characterization, we have secondly introduced the notion of asynchronous iteration graph recalled bellow.
Let $f$ be a map from $\mathds{B}^n$ to itself. The
{\emph{asynchronous iteration graph}} associated with $f$ is the
if and only if $\Gamma(f)$ is strongly connected.
\end{theorem}
-
-
+This result of chaos has lead us to study the possibility to build a pseudo-random number generator (PRNG) based on the chaotic iterations.
+As $G_f$, defined on the domain $\llbracket 1 ; n \rrbracket^{\mathds{N}} \times \mathds{B}^n$, is build from Boolean networks $f : \mathds{B}^n \rightarrow \mathds{B}^n$, we can preserve the theoretical properties on $G_f$ during implementations (due to the discrete nature of $f$). It is as if $\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ; n \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
\section{Application to Pseudo-Randomness}
\end{theorem}
+
+\alert{Mettre encore un peu de blabla sur le PRNG, puis enchaîner en disant que, ok, on peut préserver le chaos quand on passe sur machine, mais que le chaos dont il s'agit a été prouvé pour une distance bizarroïde sur un espace non moins hémoroïde, d'où ce qui suit}
+
+
+
\section{The relativity of disorder}
\label{sec:de la relativité du désordre}
