X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/b5e3d39489874c8ac95b5825ed5baa520529de17..016edf5f7ca01fdf537b4abcb6de87d75c604510:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 537feef..57526f2 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -48,9 +48,10 @@ This is the abstract
 
 Interet des itÃ©rations chaotiques pour gÃ©nÃ©rer des nombre alea\\
 Interet de gÃ©nÃ©rer 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.
@@ -191,11 +192,10 @@ The distance presented above follows these recommendations. Indeed, if the floor
 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
@@ -216,8 +216,8 @@ Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic  (according to  Devaney)
 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}
 
@@ -285,6 +285,11 @@ We have proven in \cite{FCT11} that,
 \end{theorem} 
 
 
+
+\alert{Mettre encore un peu de blabla sur le PRNG, puis enchaÃ®ner en disant que, ok, on peut prÃ©server le chaos quand on passe sur machine, mais que le chaos dont il s'agit a Ã©tÃ© prouvÃ© pour une distance bizarroÃ¯de sur un espace non moins hÃ©moroÃ¯de, d'oÃ¹ ce qui suit}
+
+
+
 \section{The relativity of disorder}
 \label{sec:de la relativitÃ© du dÃ©sordre}
 
