-Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$,
+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 $f:\Bool^{\mathsf{N}} \rightarrow
-\Bool^\mathsf{N}$,
+\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 $x^0$.
\begin{algorithm}[ht]
%\begin{scriptsize}
-\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)}
-\KwOut{a configuration $x$ ($n$ bits)}
+\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{$\textit{Random}(1) \neq 0$}{
-$s\leftarrow{\textit{Random}(n)}$\;
-$x\leftarrow{F_f(s,x)}$\;
+$s^0\leftarrow{\textit{Random}(\mathsf{N})}$\;
+$x\leftarrow{F_f(x,s^0)}$\;
}
}
return $x$\;
%\end{scriptsize}
-\caption{Pseudo Code of the $\chi_{\textit{15Rairo}}$ PRNG}
+\caption{Pseudo Code of the $\chi_{\textit{16HamG}}$ PRNG}
\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 $\chi_{\textit{15Rairo}}$ algorithm
+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}.
+given in Sect.~\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}))
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 $E[\tau]$)
+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}).
+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|}
+\begin{tabular}{|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\\
+$\textcircled{a}$&[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 & 236\\
+[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&335\\
+&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, 90, 113, 88, 115, 126, 125, 84, 123, 98,
+&&\\
+&112, 96, 109, 106, 77, 110, 99, 74, 104, 72, 71, 100, 117, 116,
+&&\\
+&103, 102, 65, 97, 31, 86, 95, 28, 27, 91, 121, 92, 119, 118, 69,
+&&\\
+&68, 87, 114, 89, 81, 15, 76, 79, 108, 107, 10, 105, 8, 7, 6, 101,
+&&\\
+$\textcircled{d}$&70, 75, 82, 64, 0, 127, 54, 53, 62, 51, 59, 56, 60, 39, 52, 37,
+&&\\
+&36, 55, 58, 57, 49, 63, 44, 47, 40, 42, 46, 45, 41, 35, 34, 33,
+&&\\
+&38, 43, 50, 32, 48, 29, 94, 61, 24, 26, 18, 17, 25, 19, 30, 85,
+&&\\
+&22, 83, 2, 16, 80, 13, 78, 9, 14, 3, 11, 73, 12, 23, 4, 21, 20,
+&&\\
+&67, 66, 5, 1]
+
+
+
+
+
+
+
[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\\
+&7 & 99\\
&
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\\
+&8&110\\
&
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}
