From ac6bb3190276a11caaed94b7443f95f697328bcd Mon Sep 17 00:00:00 2001
From: =?utf8?q?Jean-Fran=C3=A7ois=20Couchot?=
 <couchot@couchot.iut-bm.univ-fcomte.fr>
Date: Fri, 13 Mar 2015 10:04:30 +0100
Subject: [PATCH] generating : synthese secrypt + mons

---
 chaos.tex    |  6 ++++++
 main.tex     | 27 ++++++++++++---------------
 stopping.tex |  8 +++++++-
 3 files changed, 25 insertions(+), 16 deletions(-)

diff --git a/chaos.tex b/chaos.tex
index 91b8717..569d44a 100644
--- a/chaos.tex
+++ b/chaos.tex
@@ -524,3 +524,9 @@ and only if its iteration graph $\Gamma_{\mathcal{P}}(f)$ is strongly connected.
   If $b$ is even, any vertex $e$ of $\Gamma_{\{b\}}(f_0)$ cannot reach itself 
   and thus $\Gamma_{\{b\}}(f_0)$ is not strongly connected.
 \end{proof}
+
+The next section shows how to generate functions and a iteration number $b$
+such that $\Gamma_{\{b\}}$ is strongly connected.
+
+
+ 
\ No newline at end of file
diff --git a/main.tex b/main.tex
index bc8524b..b0f9c8a 100644
--- a/main.tex
+++ b/main.tex
@@ -86,7 +86,9 @@
 
 
 
-\title{XXX}
+\title{Random Walk in a N-cube Without Hamiltonian Cycle 
+  to Chaotic Pseudorandom Number Generation: Theoretical and Practical 
+  Considerations}
 
 \begin{document}
 
@@ -96,15 +98,13 @@
 
 
 \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 desgin 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
@@ -116,13 +116,10 @@ may be  updated at each iteration.   At the theoretical level,  we show that
 \input{preliminaries}
 
 \section{Proof Of Chaos}
-\JFC{Enlever les refs aux PRNGs, harmoniser l'exemple}
 \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 $\Gamma_{\{b\}}(f)$}
+\input{generating}
 
 \section{Random walk on the modified Hypercube}
 \input{stopping}
diff --git a/stopping.tex b/stopping.tex
index 96fd41b..0ad3e8a 100644
--- a/stopping.tex
+++ b/stopping.tex
@@ -345,4 +345,10 @@ Theorem~\ref{prop:stop} is a direct application of
 lemma~\ref{prop:lambda} and~\ref{lm:stopprime}.
 \end{proof}
 
-
+Notice that the calculus of the stationary time upper bound is obtained
+under the following constraint: for each vertex in the $\mathsf{N}$-cube 
+there are one ongoing arc and one outgoing arc that are removed. 
+The calculus does not consider (balanced) hamiltonian cycles, which 
+are more regular and more binding than this constraint.
+In this later context, we claim that the upper bound for the stopping time 
+should be reduced.
-- 
2.39.5