X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/586298b57c6e1cec5566bbe692ac21c4c701fd51..70b2c7214ae3421e58d225cbbb21fafbebcd8acc:/main.tex?ds=sidebyside

diff --git a/main.tex b/main.tex
index 340e8d3..9e544ca 100644
--- a/main.tex
+++ b/main.tex
@@ -1,14 +1,18 @@
 \documentclass{ita}
 \usepackage{graphicx}
+\usepackage{caption}
+\usepackage{subcaption}
+
 \usepackage{dsfont}
 \usepackage{stmaryrd}
-\usepackage[font=footnotesize]{subfig}
+%\usepackage[font=footnotesize]{subfig}
 \usepackage{ifthen}
 \usepackage{color}
 \usepackage{algorithm2e}
+\usepackage{epstopdf}
+%\usepackage{ntheorem}
 
-
-\usepackage[latin1]{inputenc}
+\usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc} 
 \usepackage[english]{babel}
 \usepackage{amsmath,amssymb,latexsym,eufrak,euscript}
@@ -27,7 +31,7 @@
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Définitions personnelles
+% Définitions personnelles
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \definecolor{bleuclair}{rgb}{0.75,0.75,1.0}
@@ -54,22 +58,13 @@
 \def \ts {\tau_{\rm stop}}
 
 
-% \theoremstyle{plain}
-% \theoremheaderfont{\normalfont\bfseries\sc}
-% \theorembodyfont{\slshape}
-% \theoremsymbol{\ensuremath{\diamondsuit}}
-% \theoremprework{\bigskip}
-% \theoremseparator{.}
-\newtheorem{Def}{\underline{Définition}}
-\newtheorem{Lemma}{\underline{Lemme}}
-\newtheorem{Theo}{\underline{Théorème}}
-% \theoremheaderfont{\sc}
-% \theorembodyfont{\upshape}
-% \theoremstyle{nonumberplain}
-% \theoremseparator{}
-% \theoremsymbol{\rule{1ex}{1ex}}
-\newtheorem{Proof}{Preuve :}
-\newtheorem{xpl}{Exemple illustratif :}
+\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}}
 %\newcommand{\ie}{\textit{i.e.}}
@@ -81,7 +76,7 @@
 \newcommand{\Gall}[0]{\ensuremath{\mathcal{G}}}
 
 
-\newcommand{\JFC}[1]{\begin{color}{green}\textit{}\end{color}}
+\newcommand{\JFC}[1]{\begin{color}{green}\textit{#1}\end{color}}
 \newcommand{\CG}[1]{\begin{color}{blue}\textit{}\end{color}}
 \newcommand{\og}[0]{``}
 \newcommand{\fg}[1]{''}
@@ -91,60 +86,66 @@
 
 
 
-\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-Cyrile Heam}
+\address{Institut FEMTO-ST, Université de Franche-Comté, Belfort, France}
 
-\author{Sylvain Contassot-Vivier}
-\address{Loria - UMR 7503, Université de Lorraine, Nancy, France}
 
-\date{...}
 
 \begin{abstract}
-This   paper  extends   the  results   presented  in~\cite{bcgr11ip}
-and~\cite{chgw14oip} 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 passe
+the classical statsitcal tests.
 \end{abstract}
 
+\maketitle
+
 \section{Introduction}
-%\input{intro}
+\input{intro}
 
  \section{\uppercase{Preliminaries}}\label{sec:preliminaries}
 \input{preliminaries}
 
-\section{Stopping Time}
-\input{stopping}
+\section{Proof Of Chaos}\label{sec:proofOfChaos}
+\input{chaos}
+
+\section{Functions with Strongly Connected $\Gamma_{\{b\}}(f)$}\label{sec:SCCfunc}
+\input{generating}
 
-\section{Jumping in a specific $n$-cube}
-% Proposer alors les sauts dans ce n-cube (nouveau)
-%  Il y a des preuves que j'ai faites dans TSI sur la préservation des propriétés de chaos lorsqu'on saute, qu'on peut traduire et qui n'ont pas été publiées en anglais, je crois (nouveau). 
+\section{Random walk on the modified Hypercube}\label{sec:hypercube}
+\input{stopping}
 
-\section{Stopping Time (continued)}
 % 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). 
+% 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{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{Expérimentations}
-%
+\section{Application to Pseudorandom Number Generation}\label{sec:prng}
+\input{prng}
+\JFC{ajouter ici les expérimentations}
+
 
 \section{Conclusion}
-%\input{conclusion}
+\input{conclusion}
 
 %\acknowledgements{...}
 
-
+\bibliographystyle{alpha}
 \bibliography{biblio}
 
 \end{document}