X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/5e3dba6ade7cdb2d622eec1fd7d7ef5a0a168b4d..2644c7bc8564e7829d6f0d9b2e2a66e6460aec67:/prng.tex?ds=sidebyside

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.