From 2644c7bc8564e7829d6f0d9b2e2a66e6460aec67 Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Sat, 14 Mar 2015 15:16:01 +0100
Subject: [PATCH] debut d'experimentations

---
 main.tex     |  2 +-
 prng.tex     | 30 ++++++++++++++++--------------
 stopping.tex |  6 +++---
 3 files changed, 20 insertions(+), 18 deletions(-)

diff --git a/main.tex b/main.tex
index 9e544ca..92832e5 100644
--- a/main.tex
+++ b/main.tex
@@ -135,7 +135,7 @@ the classical statsitcal tests.
 %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}\label{sec:prng}
+\section{Experiments}\label{sec:prng}
 \input{prng}
 \JFC{ajouter ici les expérimentations}
 
diff --git a/prng.tex b/prng.tex
index 88253cf..45fcfcd 100644
--- a/prng.tex
+++ b/prng.tex
@@ -1,6 +1,7 @@
 Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$
-which is based on random walks in $\Gamma(f)$. 
-More precisely, let be given a Boolean map $f:\Bool^n \rightarrow \Bool^n$,
+which is based on random walks in $\Gamma_{\{b\}}(f)$. 
+More precisely, let be given a Boolean map $f:\Bool^{\mathsf{N}} \rightarrow 
+\Bool^\mathsf{N}$,
 a PRNG \textit{Random},
 an integer $b$ that corresponds an iteration number (\textit{i.e.}, the length of the walk), and 
 an initial configuration $x^0$. 
@@ -32,21 +33,22 @@ return $x$\;
 \end{algorithm}
 
 
-This PRNG is a particularized version of Algorithm given in~\cite{DBLP:conf/secrypt/CouchotHGWB14}.
+This PRNG is slightly different from $\chi_{\textit{14Secrypt}}$
+recalled in Algorithm~\ref{CI Algorithm}.
 As this latter, the length of the random 
 walk of our algorithm is always constant (and is equal to $b$). 
 However, in the current version, we add the constraint that   
 the probability to execute the function $F_f$ is equal to 0.5 since
 the output of $\textit{Random(1)}$ is uniform in $\{0,1\}$.  
+This constraint is added to match the theoretical framework of 
+Sect.~\ref{sec:hypercube}.
+
+
+
+Notice that the chaos property of $G_f$ given in Sect.\ref{sec:proofOfChaos}
+only requires that the graph $\Gamma_{\{b\}}(f)$ is strongly connected.
+Since the $\chi_{\textit{15Rairo}}$ algorithme 
+only adds propbability constraints on existing edges, 
+it preserves this property. 
+
 
-Let $f: \Bool^{n} \rightarrow \Bool^{n}$.
-It has been shown~\cite[Th. 4, p. 135]{bcgr11:ip} that 
-if its iteration graph is strongly connected, then 
-the output of $\chi_{\textit{14Secrypt}}$ follows 
-a law that tends to the uniform distribution 
-if and only if its Markov matrix is a doubly stochastic matrix.
-  
-Let us now present  a method to
-generate  functions
-with Doubly Stochastic matrix and Strongly Connected iteration graph,
-denoted as DSSC matrix.   
diff --git a/stopping.tex b/stopping.tex
index e413688..eec08aa 100644
--- a/stopping.tex
+++ b/stopping.tex
@@ -289,10 +289,10 @@ More formally,
 since $\ov{h}$ is square-free,
 $\ov{h}(X)=\ov{h}(\ov{h}(\ov{h}^{-1}(X)))\neq \ov{h}^{-1}(X)$. It follows
 that $(X,\ov{h}^{-1}(X))\in E_h$. We thus have
-$P(X_1=\ov{h}^{-1}(X))=\frac{1}{2{\MATHSF{N}}}$. Now, by Lemma~\ref{lm:h},
+$P(X_1=\ov{h}^{-1}(X))=\frac{1}{2{\mathsf{N}}}$. Now, by Lemma~\ref{lm:h},
 $h(\ov{h}^{-1}(X))\neq h(X)$. Therefore $\P(S_{x,\ell}=2\mid
-X_1=\ov{h}^{-1}(X))=\frac{1}{2{\MATHSF{N}}}$, proving that $\P(S_{x,\ell}\leq 2)\geq
-\frac{1}{4{\MATHSF{N}}^2}$.
+X_1=\ov{h}^{-1}(X))=\frac{1}{2{\mathsf{N}}}$, proving that $\P(S_{x,\ell}\leq 2)\geq
+\frac{1}{4{\mathsf{N}}^2}$.
 \end{itemize}
 
 
-- 
2.39.5