\def \ts {\tau_{\rm stop}}
-\newtheorem{Def}{Definition}
-%\newtheorem{Lemma}{\underline{Lemma}}
-\newtheorem{Theo}{Theorem}
-\newtheorem{Corollary}{Corollary}
-\newtheorem{Lemma}{Lemma}
-\newtheorem{proposition}{Proposition}
\newtheorem*{xpl}{Running Example}
\newcommand{\vectornorm}[1]{\ensuremath{\left|\left|#1\right|\right|_2}}
-\title{XXX}
+\title{Random Walk in a N-cube Without Hamiltonian Cycle
+ to Chaotic Pseudorandom Number Generation: Theoretical and Practical
+ Considerations}
\begin{document}
-\author{Jean-François Couchot, Christophe Guyeux, Pierre-Cyrile Heam}
-\address{Institut FEMTO-ST, Université de Franche-Comté, Belfort, France}
+\author{Jean-François Couchot, Christophe Guyeux, Pierre-Cyrille Heam}
+\address{FEMTO-ST Institute, University of Franche-Comté, Belfort, France}
+\keywords{Pseudorandom Number Generator, Theory of Chaos, Markov Matrice, Hamiltonian Path, Mixing Time, Stopping Time, Statistical Test}
+\subjclass{34C28, 37A25,11K45}
\begin{abstract}
-This paper extends the results presented in~\cite{bcgr11:ip}
-and~\cite{DBLP:conf/secrypt/CouchotHGWB14}
-by using the \emph{chaotic} updating mode, in the sense
-of F. Robert~\cite{Robert}. In this mode, several components of the system
-may be updated at each iteration. At the theoretical level, we show that
- the properties of chaos and uniformity of the obtained PRNG are preserved.
- At the practical level, we show that the algorithm that builds strongly
- connected iteration graphs, with doubly stochastic Markov matrix, has a
- reduced mixing time.
+This paper is dedicated to the design of chaotic random generators
+and extends previous works proposed by some of the authors.
+We propose a theoretical framework proving both the chaotic properties and
+that the limit distribution is uniform.
+A theoretical bound on the stationary time is given and
+practical experiments show that the generators successfully pass
+the classical statistical tests.
\end{abstract}
\maketitle
\section{Introduction}
-%\input{intro}
+\input{intro}
\section{\uppercase{Preliminaries}}\label{sec:preliminaries}
\input{preliminaries}
-\section{Proof Of Chaos}
-\JFC{Enlever les refs aux PRNGs, harmoniser l'exemple}
+\section{Proof Of Chaos}\label{sec:proofOfChaos}
\input{chaos}
-\section{Generating....}
-\JFC{Reprendre Mons en synthétisant... conclusion: n-cube moins hamitonien.
-question efficacité d'un tel algo}
-%\input{chaos}
+\section{Functions with Strongly Connected $\Gamma_{\{b\}}(f)$}\label{sec:SCCfunc}
+\input{generating}
-\section{Random walk on the modified Hypercube}
+\section{Stopping Time}\label{sec:hypercube}
\input{stopping}
-% Donner la borne du stopping time quand on marche dedans (nouveau).
-% Énoncer le problème de la taille de cette borne
-% (elle est certes finie, mais grande).
-
-
-
-
-%\section{Quality study of the strategy}
-%6) Se pose alors la question de comment générer une stratégie de "bonne qualité". Par exemple, combien de générateurs aléatoires embarquer ? (nouveau)
-
-
-\section{Application to Pseudorandom Number Generation}
+\section{Experiments}\label{sec:prng}
\input{prng}
-\JFC{ajouter ici les expérimentations}
+
\section{Conclusion}
-%\input{conclusion}
+\input{conclusion}
%\acknowledgements{...}
