-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}.
return $x$\;
%\end{scriptsize}
\caption{Pseudo Code of the $\chi_{\textit{15Rairo}}$ PRNG}
-\label{CI Algorithm}
+\label{CI Algorithm:2}
\end{algorithm}
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.
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}).
&$E[\tau]$\\
\hline
%%%%% n= 4
-$\textcircled{a}$&[13,10,9,14,3,11,1,12,15,4,7,5,2,6,0,8]&4&64&\\
+$\textcircled{a}$&[13,10,9,14,3,11,1,12,15,4,7,5,2,6,0,8]&4&64&154\\
\hline
%%%%% n= 5
$\textcircled{b}$&
-[29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, & 5 & 78 & \\
+[29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, & 5 & 78 & 236\\
&
31, 12, 15, 8, 10, 14, 13, 9, 3, 2, 7, 20, 11, 18, 0, 4]
&&&\\
&&&\\
$\textcircled{c}$&
26, 25, 30, 19, 27, 17, 28, 31, 20, 23, 21, 18, 22, 16, 24, 13,
-&6&88&\\
+&6&88&335\\
&
12, 29, 8, 43, 14, 41, 0, 5, 38, 4, 6, 11, 3, 9, 32]
&&&\\
&&&\\
$\textcircled{d}$&
69, 20, 19, 114, 17, 112, 77, 76, 13, 108, 74, 10, 9, 73, 67, 66,
-&7 & 99&\\
+&7 & 99&450\\
&
101, 100, 75, 82, 97, 0, 127, 54, 57, 62, 51, 59, 56, 48, 53, 38,
&&&\\
$\textcircled{e}$&
8, 7, 198, 197, 4, 195, 2, 161, 160, 255, 124, 109, 108, 122,
-&8&110&\\
+&8&110&582\\
&
126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106,
&&&\\
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}
\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 ($\mathbb{P}_T$)}
