X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/9fc003099dc86caaa2ccf0645be2764c81418534..b14071948f5418eda195be08e12edc746770f7de:/prng.tex?ds=sidebyside

diff --git a/prng.tex b/prng.tex
index 0667cca..3776d09 100644
--- a/prng.tex
+++ b/prng.tex
@@ -1,13 +1,13 @@
-Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$
+Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$,
 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 integer $b$ that corresponds to an iteration number (\textit{i.e.}, the length of the walk), and 
 an initial configuration $x^0$. 
 Starting from $x^0$, the algorithm repeats $b$ times 
-a random choice of which edge to follow and traverses this edge 
-provided it is allowed to traverse it, \textit{i.e.}, 
+a random choice of which edge to follow, and traverses this edge 
+provided it is allowed to do so, \textit{i.e.}, 
 when $\textit{Random}(1)$ is not null. 
 The final configuration is thus outputted.
 This PRNG is formalized in Algorithm~\ref{CI Algorithm}.
@@ -47,14 +47,14 @@ 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, 
+Since the $\chi_{\textit{15Rairo}}$ algorithm 
+only adds probability constraints on existing edges, 
 it preserves this property. 
 
 
 For each number $\mathsf{N}=4,5,6,7,8$ of bits, we have generated 
-the functions according the method 
-given in Sect.~\ref{sec:SCCfunc} .
+the functions according to the method 
+given in Sect.~\ref{sec:SCCfunc}.
 For each $\mathsf{N}$, we have then restricted this evaluation to the function 
 whose Markov Matrix (issued from Eq.~(\ref{eq:Markov:rairo})) 
 has the smallest practical mixing time.
@@ -69,10 +69,10 @@ it is obtained as  the  binary  value  of  the  fourth element  in
 the  second  list (namely~14).  
 
 In this table the column 
-which is labeled with $b$ (respectively by $E[\tau]$)
+that is labeled with $b$ (respectively by $E[\tau]$)
 gives the practical mixing time 
-where the deviation to the standard distribution is less than $10^{-6}$
-(resp. the theoretical upper bound ofstopping time as described in 
+where the deviation to the standard distribution is lesser than $10^{-6}$
+(resp. the theoretical upper bound of stopping time as described in 
 Sect.~\ref{sec:hypercube}).
 
 
@@ -214,7 +214,9 @@ If the value $\mathbb{P}_T$ of any test is smaller than 0.0001, the sequences ar
 and the generator is unsuitable. Table~\ref{The passing rate} shows $\mathbb{P}_T$ of sequences based on discrete
 chaotic iterations using different schemes. If there are at least two statistical values in a test, this test is
 marked with an asterisk and the average value is computed to characterize the statistics.
-We can see in Table \ref{The passing rate} that all the rates are greater than 97/100, \textit{i. e.}, all the generators pass the NIST test.
+We can see in Table \ref{The passing rate} that all the rates are greater than 97/100, \textit{i.e.}, all the generators 
+achieve to pass the NIST battery of tests.
+
 
 
 \begin{table}