-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 $f:\Bool^n \rightarrow \Bool^n$,
+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}
-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 $\chi_{\textit{15Rairo}}$ 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 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.
+Such functions are
+given in Table~\ref{table:nc}.
+In this table, let us consider for instance
+the function $\textcircled{a}$ from $\Bool^4$ to $\Bool^4$
+defined by the following images :
+$[13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]$.
+In other words, the image of $3~(0011)$ by $\textcircled{a}$ is $14~(1110)$:
+it is obtained as the binary value of the fourth element in
+the second list (namely~14).
+
+In this table the column
+that is labeled with $b$ (respectively by $E[\tau]$)
+gives the practical mixing time
+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}).
+
+
+
+\begin{table*}[t]
+\begin{center}
+\begin{scriptsize}
+\begin{tabular}{|c|c|c|c|c|}
+\hline
+Function $f$ & $f(x)$, for $x$ in $(0,1,2,\hdots,2^n-1)$ & $\mathsf{N}$ & $b$
+&$E[\tau]$\\
+\hline
+%%%%% n= 4
+$\textcircled{a}$&[13,10,9,14,3,11,1,12,15,4,7,5,2,6,0,8]&4&64&154\\
+\hline
+%%%%% n= 5
+$\textcircled{b}$&
+[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]
+&&&\\
+%%%%% n= 6
+\hline
+&
+[55, 60, 45, 44, 58, 62, 61, 48, 53, 50, 52, 36, 59, 34, 33, 49,
+&&&\\
+&
+ 15, 42, 47, 46, 35, 10, 57, 56, 7, 54, 39, 37, 51, 2, 1, 40, 63,
+&&&\\
+$\textcircled{c}$&
+ 26, 25, 30, 19, 27, 17, 28, 31, 20, 23, 21, 18, 22, 16, 24, 13,
+&6&88&335\\
+&
+12, 29, 8, 43, 14, 41, 0, 5, 38, 4, 6, 11, 3, 9, 32]
+&&&\\
+%%%%% n= 7
+\hline
+&
+[111, 94, 93, 116, 122, 90, 125, 88, 115, 126, 119, 84, 123, 98,
+&&&\\
+&
+ 81, 120, 109, 106, 105, 110, 99, 107, 104, 72, 71, 118, 117,
+ &&&\\
+&
+96, 103, 102, 113, 64, 79, 86, 95, 124, 83, 91, 121, 24, 85, 22,
+&&&\\
+$\textcircled{d}$&
+69, 20, 19, 114, 17, 112, 77, 76, 13, 108, 74, 10, 9, 73, 67, 66,
+&7 & 99&450\\
+
+&
+ 101, 100, 75, 82, 97, 0, 127, 54, 57, 62, 51, 59, 56, 48, 53, 38,
+&&&\\
+&
+ 37, 60, 55, 58, 33, 49, 63, 44, 47, 40, 42, 46, 45, 41, 35, 34,
+&&&\\
+&
+39, 52, 43, 50, 32, 36, 29, 28, 61, 92, 26, 18, 89, 25, 87, 30,
+&&&\\
+&
+23, 4, 27, 2, 16, 80, 31, 78, 15, 14, 3, 11, 8, 12, 5, 70, 21,
+&&&\\
+&
+68, 7, 6, 65, 1]
+&&&\\
+
+
+%%%%%n=8
+\hline
+&
+[223, 190, 249, 254, 187, 251, 233, 232, 183, 230, 247, 180, 227,
+&&&\\
+&
+178, 240, 248, 237, 236, 253, 172, 203, 170, 201, 168, 229, 166,
+&&&\\
+&
+165, 244, 163, 242, 241, 192, 215, 220, 205, 216, 218, 222, 221,
+&&&\\
+&
+208, 213, 210, 212, 214, 219, 211, 217, 209, 239, 202, 207, 140,
+&&&\\
+&
+139, 234, 193, 204, 135, 196, 199, 132, 194, 130, 225, 200, 159,
+&&&\\
+&
+62, 185, 252, 59, 250, 169, 56, 191, 246, 245, 52, 243, 50, 176,
+&&&\\
+&
+48, 173, 238, 189, 44, 235, 42, 137, 184, 231, 38, 37, 228, 35,
+&&&\\
+&
+226, 177, 224, 151, 156, 141, 152, 154, 158, 157, 144, 149, 146,
+&&&\\
+&
+148, 150, 155, 147, 153, 145, 175, 206, 143, 136, 11, 142, 129,
+&&&\\
+$\textcircled{e}$&
+8, 7, 198, 197, 4, 195, 2, 161, 160, 255, 124, 109, 108, 122,
+&8&110&582\\
+&
+ 126, 125, 112, 117, 114, 116, 100, 123, 98, 97, 113, 79, 106,
+&&&\\
+&
+ 111, 110, 99, 74, 121, 120, 71, 118, 103, 101, 115, 66, 65,
+&&&\\
+&
+104, 127, 90, 89, 94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86,
+&&&\\
+&
+80, 88, 77, 76, 93, 72, 107, 78, 105, 64, 69, 102, 68, 70, 75,
+&&&\\
+&
+67, 73, 96, 55, 58, 45, 188, 51, 186, 61, 40, 119, 182, 181,
+&&&\\
+&
+53, 179, 54, 33, 49, 15, 174, 47, 60, 171, 46, 57, 32, 167, 6,
+&&&\\
+&
+ 36, 164, 43, 162, 1, 0, 63, 26, 25, 30, 19, 27, 17, 28, 31,
+&&&\\
+&
+20, 23, 21, 18, 22, 16, 24, 13, 10, 29, 14, 3, 138, 41, 12,
+&&&\\
+&
+39, 134, 133, 5, 131, 34, 9, 128]
+&&&\\
+\hline
+\end{tabular}
+\end{scriptsize}
+\end{center}
+\caption{Functions with DSCC Matrix and smallest MT\label{table:nc}}
+\end{table*}
+
+
+
+Let us first discuss about results against the NIST test suite.
+In our experiments, 100 sequences (s = 100) of 1,000,000 bits are generated and tested.
+If the value $\mathbb{P}_T$ of any test is smaller than 0.0001, the sequences are considered to be not good enough
+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
+achieve to pass the NIST battery of tests.
+
+
+
+\begin{table}
+\renewcommand{\arraystretch}{1.3}
+\begin{center}
+\begin{scriptsize}
+\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 ($\mathbb{P}_T$)}
+\label{The passing rate}
+\end{table}
+
-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.
