Let us finally present the pseudorandom number generator $\chi_{\textit{16HamG}}$,
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 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 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:2}.



\begin{algorithm}[ht]
%\begin{scriptsize}
\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($\mathsf{N}$ bits)}
\KwOut{a configuration $x$ ($\mathsf{N}$ bits)}
$x\leftarrow x^0$\;
\For{$i=0,\dots,b-1$}
{
\If{$\textit{Random}(1) \neq 0$}{
$s^0\leftarrow{\textit{Random}(\mathsf{N})}$\;
$x\leftarrow{F_f(x,s^0)}$\;
}
}
return $x$\;
%\end{scriptsize}
\caption{Pseudo Code of the $\chi_{\textit{16HamG}}$ PRNG}
\label{CI Algorithm:2}
\end{algorithm}


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 $\chi_{\textit{16HamG}}$ 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} and~\ref{sec:hamilton}. 
% MENTION FILTRAGE POSSIBLE LORS DE CONSTRUCTION... (SCV) 
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|}
\hline
Function $f$ & $f(x)$, for $x$ in $(0,1,2,\hdots,2^n-1)$ & $\mathsf{N}$ & $b$ 
\\ 
\hline
%%%%% n= 4
$\textcircled{a}$&[13,10,9,14,3,11,1,12,15,4,7,5,2,6,0,8]&4&64\\
\hline
%%%%% n= 5
$\textcircled{b}$& 
[29, 22, 25, 30, 19, 27, 24, 16, 21, 6, 5, 28, 23, 26, 1, 17, & 5 & 78 \\
&
 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\\
&
12, 29, 8, 43, 14, 41, 0, 5, 38, 4, 6, 11, 3, 9, 32]
&&\\
%%%%% n= 7
\hline
&
[111, 124, 93, 120, 122, 114, 89, 121, 87, 126, 125, 84, 123, 82, 
&&\\
&112, 80, 79, 106, 105, 110, 75, 107, 73, 108, 119, 100, 117, 116, 
&&\\
&103, 102, 101, 97, 31, 86, 95, 94, 83, 26, 88, 24, 71, 118, 69, 
&&\\
&68, 115, 90, 113, 16, 15, 76, 109, 72, 74, 10, 9, 104, 7, 6, 65, 
&&\\
$\textcircled{d}$ &70, 99, 98, 64, 96, 127, 54, 53, 62, 51, 59, 56, 60, 39, 52, 37, &7 &99\\
&36, 55, 58, 57, 49, 63, 44, 47, 40, 42, 46, 45, 41, 35, 34, 33, 
&&\\
&38, 43, 50, 32, 48, 29, 28, 61, 92, 91, 18, 17, 25, 19, 30, 85, 
&&\\
&22, 27, 2, 81, 0, 13, 78, 77, 14, 3, 11, 8, 12, 23, 4, 21, 20, 
&&\\
&67, 66, 5, 1]
&&\\


%%%%%n=8
\hline
&
[223, 238, 249, 254, 243, 251, 233, 252, 183, 244, 229, 245, 227, 
&&\\
&246, 240, 176, 175, 174, 253, 204, 203, 170, 169, 248, 247, 226, 
&&\\
&228, 164, 163, 162, 161, 192, 215, 220, 205, 216, 155, 222, 221, 
&&\\
&208, 213, 150, 212, 214, 219, 211, 145, 209, 239, 202, 207, 140, 
&&\\
&195, 234, 193, 136, 231, 230, 199, 197, 131, 198, 225, 200, 63, 
&&\\
&188, 173, 184, 186, 250, 57, 168, 191, 178, 180, 52, 187, 242, 
&&\\
&241, 48, 143, 46, 237, 236, 235, 138, 185, 232, 135, 38, 181, 165, 
&&\\
&35, 166, 33, 224, 31, 30, 153, 158, 147, 218, 217, 156, 159, 148, 
&&\\
$\textcircled{e}$&151, 149, 19, 210, 144, 152, 141, 206, 13, 12, 171, 10, 201, 128, 
&8&109\\
&133, 130, 132, 196, 3, 194, 137, 0, 255, 124, 109, 120, 122, 106, 
&&\\
&125, 104, 103, 114, 116, 118, 123, 98, 97, 113, 79, 126, 111, 110, 
&&\\
&99, 74, 121, 72, 71, 70, 117, 101, 115, 102, 65, 112, 127, 90, 89, 
&&\\
&94, 83, 91, 81, 92, 95, 84, 87, 85, 82, 86, 80, 88, 77, 76, 93, 
&&\\
&108, 107, 78, 105, 64, 69, 66, 68, 100, 75, 67, 73, 96, 55, 190, 
&&\\
&189, 62, 51, 59, 41, 60, 119, 182, 37, 53, 179, 54, 177, 32, 45, 
&&\\
&44, 61, 172, 11, 58, 9, 56, 167, 34, 36, 4, 43, 50, 49, 160, 23, 
&&\\
&28, 157, 24, 26, 154, 29, 16, 21, 18, 20, 22, 27, 146, 25, 17, 47, 
&&\\
&142, 15, 14, 139, 42, 1, 40, 39, 134, 7, 5, 2, 6, 129, 8]
&&\\
\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{tiny}
\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 
\begin{tabular}{|l|r|r|r|r|r||r|r|r|r|r|}
 \hline 
Test & $\textit{MT}_4$ & $\textit{MT}_5$& $\textit{MT}_6$& $\textit{MT}_7$& $\textit{MT}_8$
&$\textcircled{a}$& $\textcircled{b}$ & $\textcircled{c}$ & $\textcircled{d}$ & $\textcircled{e}$ \\ \hline 
Frequency (Monobit)& 0.924 (1.0)& 0.678 (0.98)& 0.102 (0.97)& 0.213 (0.98)& 0.719 (0.99)& 0.129 (1.0)& 0.181 (1.0)& 0.637 (0.99)& 0.935 (1.0)& 0.978 (1.0)\\ \hline 
Frequency  within a Block& 0.514 (1.0)& 0.419 (0.98)& 0.129 (0.98)& 0.275 (0.99)& 0.455 (0.99)& 0.275 (1.0)& 0.534 (0.98)& 0.066 (1.0)& 0.719 (1.0)& 0.366 (1.0)\\ \hline 
Cumulative Sums (Cusum) *& 0.668 (1.0)& 0.568 (0.99)& 0.881 (0.98)& 0.529 (0.98)& 0.657 (0.995)& 0.695 (1.0)& 0.540 (1.0)& 0.514 (0.985)& 0.773 (0.995)& 0.506 (0.99)\\ \hline 
Runs& 0.494 (0.99)& 0.595 (0.97)& 0.071 (0.97)& 0.017 (1.0)& 0.834 (1.0)& 0.897 (0.99)& 0.051 (1.0)& 0.102 (0.98)& 0.616 (0.99)& 0.191 (1.0)\\ \hline 
Longest Run of Ones in a Block& 0.366 (0.99)& 0.554 (1.0)& 0.042 (0.99)& 0.051 (0.99)& 0.897 (0.97)& 0.851 (1.0)& 0.595 (0.99)& 0.419 (0.98)& 0.616 (0.98)& 0.897 (1.0)\\ \hline 
Binary Matrix Rank& 0.275 (0.98)& 0.494 (0.99)& 0.719 (1.0)& 0.334 (0.98)& 0.637 (0.99)& 0.419 (1.0)& 0.946 (0.99)& 0.319 (0.99)& 0.739 (0.97)& 0.366 (1.0)\\ \hline 
Discrete Fourier Transform (Spectral)& 0.122 (0.98)& 0.108 (0.99)& 0.108 (1.0)& 0.514 (0.99)& 0.534 (0.98)& 0.867 (1.0)& 0.514 (1.0)& 0.145 (1.0)& 0.224 (0.99)& 0.304 (1.0)\\ \hline 
Non-overlapping Template Matching*& 0.483 (0.990)& 0.507 (0.990)& 0.520 (0.988)& 0.494 (0.988)& 0.515 (0.989)& 0.542 (0.990)& 0.512 (0.989)& 0.505 (0.990)& 0.494 (0.989)& 0.493 (0.991)\\ \hline 
Overlapping Template Matching& 0.595 (0.99)& 0.759 (1.0)& 0.637 (1.0)& 0.554 (0.99)& 0.236 (1.0)& 0.275 (0.99)& 0.080 (0.99)& 0.574 (0.98)& 0.798 (0.99)& 0.834 (0.99)\\ \hline 
Maurer's "Universal Statistical"& 0.202 (0.99)& 0.000 (0.99)& 0.514 (0.98)& 0.883 (0.97)& 0.366 (0.99)& 0.383 (0.99)& 0.991 (0.98)& 0.851 (1.0)& 0.595 (0.98)& 0.514 (1.0)\\ \hline 
Approximate Entropy (m=10)& 0.616 (0.99)& 0.145 (0.99)& 0.455 (0.99)& 0.262 (0.97)& 0.494 (1.0)& 0.935 (1.0)& 0.719 (1.0)& 0.883 (1.0)& 0.719 (0.97)& 0.366 (0.99)\\ \hline 
Random Excursions *& 0.275 (1.0)& 0.495 (0.975)& 0.465 (0.979)& 0.452 (0.991)& 0.260 (0.989)& 0.396 (0.991)& 0.217 (0.989)& 0.445 (0.975)& 0.743 (0.993)& 0.380 (0.990)\\ \hline 
Random Excursions Variant *& 0.382 (0.995)& 0.400 (0.994)& 0.417 (0.984)& 0.456 (0.991)& 0.389 (0.991)& 0.486 (0.997)& 0.373 (0.981)& 0.415 (0.994)& 0.424 (0.991)& 0.380 (0.991)\\ \hline 
Serial* (m=10)& 0.629 (0.99)& 0.963 (0.99)& 0.366 (0.995)& 0.537 (0.985)& 0.253 (0.995)& 0.350 (1.0)& 0.678 (0.995)& 0.287 (0.995)& 0.740 (0.99)& 0.301 (0.98)\\ \hline 
Linear Complexity& 0.494 (0.99)& 0.514 (0.98)& 0.145 (1.0)& 0.657 (0.98)& 0.145 (0.99)& 0.455 (0.99)& 0.867 (1.0)& 0.401 (0.99)& 0.191 (0.97)& 0.699 (1.0)\\ \hline 
\end{tabular}
\end{tiny}
\end{center}
\caption{NIST SP 800-22 test results ($\mathbb{P}_T$)}
\label{The passing rate}
\end{table*}


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