X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/d5564eebac75434c4a578ef739c24590e52b7844..5a1853fca34ff9fb85e24965d2ae79c535c16b8d:/prng.tex

diff --git a/prng.tex b/prng.tex
index db2e5e7..8b2f78f 100644
--- a/prng.tex
+++ b/prng.tex
@@ -1,16 +1,16 @@
-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}.
+This PRNG is formalized in Algorithm~\ref{CI Algorithm:2}.
 
 
 
@@ -29,7 +29,7 @@ $x\leftarrow{F_f(s,x)}$\;
 return $x$\;
 %\end{scriptsize}
 \caption{Pseudo Code of the $\chi_{\textit{15Rairo}}$ PRNG}
-\label{CI Algorithm}
+\label{CI Algorithm:2}
 \end{algorithm}
 
 
@@ -53,8 +53,8 @@ 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.
@@ -69,9 +69,9 @@ 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 $10^{-6}$
+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} 
@@ -224,6 +226,26 @@ We can see in Table \ref{The passing rate} that all the rates are greater than 9
 \setlength{\tabcolsep}{2pt}
 
 
+\begin{tabular}{|l|l|l|l|l|l|}
+\hline
+Method &$\textcircled{a}$& $\textcircled{b}$ & $\textcircled{c}$ & $\textcircled{d}$ & $\textcircled{e}$   \\ \hline\hline
+Frequency (Monobit)& 0.851 (0.98)& 0.719 (0.99)& 0.699 (0.99)& 0.514 (1.0)& 0.798 (0.99)\\ \hline 
+Frequency (Monobit)& 0.851 (0.98)& 0.719 (0.99)& 0.699 (0.99)& 0.514 (1.0)& 0.798 (0.99)\\ \hline 
+Frequency  within a Block& 0.262 (0.98)& 0.699 (0.98)& 0.867 (0.99)& 0.145 (1.0)& 0.455 (0.99)\\ \hline 
+Cumulative Sums (Cusum) *& 0.301 (0.98)& 0.521 (0.99)& 0.688 (0.99)& 0.888 (1.0)& 0.598 (1.0)\\ \hline 
+Runs& 0.224 (0.97)& 0.383 (0.97)& 0.108 (0.96)& 0.213 (0.99)& 0.616 (0.99)\\ \hline 
+Longest Run of 1s & 0.383 (1.0)& 0.474 (1.0)& 0.983 (0.99)& 0.699 (0.98)& 0.897 (0.96)\\ \hline 
+Binary Matrix Rank& 0.213 (1.0)& 0.867 (0.99)& 0.494 (0.98)& 0.162 (0.99)& 0.924 (0.99)\\ \hline 
+Disc. Fourier Transf. (Spect.)& 0.474 (1.0)& 0.739 (0.99)& 0.012 (1.0)& 0.678 (0.98)& 0.437 (0.99)\\ \hline 
+Unoverlapping Templ. Match.*& 0.505 (0.990)& 0.521 (0.990)& 0.510 (0.989)& 0.511 (0.990)& 0.499 (0.990)\\ \hline 
+Overlapping Temp. Match.& 0.574 (0.98)& 0.304 (0.99)& 0.437 (0.97)& 0.759 (0.98)& 0.275 (0.99)\\ \hline 
+Maurer's Universal Statistical& 0.759 (0.96)& 0.699 (0.97)& 0.191 (0.98)& 0.699 (1.0)& 0.798 (0.97)\\ \hline 
+Approximate Entropy (m=10)& 0.759 (0.99)& 0.162 (0.99)& 0.867 (0.99)& 0.534 (1.0)& 0.616 (0.99)\\ \hline 
+Random Excursions *& 0.666 (0.994)& 0.410 (0.962)& 0.287 (0.998)& 0.365 (0.994)& 0.480 (0.985)\\ \hline 
+Random Excursions Variant *& 0.337 (0.988)& 0.519 (0.984)& 0.549 (0.994)& 0.225 (0.995)& 0.533 (0.993)\\ \hline 
+Serial* (m=10)& 0.630 (0.99)& 0.529 (0.99)& 0.460 (0.99)& 0.302 (0.995)& 0.360 (0.985)\\ \hline 
+Linear Complexity& 0.719 (1.0)& 0.739 (0.99)& 0.759 (0.98)& 0.122 (0.97)& 0.514 (0.99)\\ \hline 
+\end{tabular}
 \end{scriptsize}
 \end{center}
 \caption{NIST SP 800-22 test results ($\mathbb{P}_T$)}