pch

[16dcc.git] / prng.tex

diff --git a/prng.tex b/prng.tex
index 5025b4eca7a9d1115ab01749ab0afe33f06660bf..8acfce7c0d98dc39de0eb523b31755b2419ecfe3 100644 (file)
--- a/prng.tex
+++ b/prng.tex
@@ -1,7 +1,7 @@
-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 
 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 $x^0$. 
 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 $x^0$. 


@@ -16,19 +16,19 @@ This PRNG is formalized in Algorithm~\ref{CI Algorithm:2}.
 
 \begin{algorithm}[ht]
 %\begin{scriptsize}
 
 \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{$\textit{Random}(1) \neq 0$}{
 $x\leftarrow x^0$\;
 \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}
 }
 }
 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}
 
 \label{CI Algorithm:2}
 \end{algorithm}
 


@@ -47,14 +47,15 @@ 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.
 
 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 
 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: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.
 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.


@@ -68,137 +69,103 @@ 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).  
 
 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 
 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{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 $(0,1,2,\hdots,2^n-1)$ & $\mathsf{N}$ & $b$ 
 \hline
 Function $f$ & $f(x)$, for $x$ in $(0,1,2,\hdots,2^n-1)$ & $\mathsf{N}$ & $b$ 

-&$E[\tau]$\\ 
+\\ 

 \hline
 %%%%% n= 4
 \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
 $\textcircled{b}$& 
 \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]
 &
  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,
 %%%%% 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,
 &
  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, 
 $\textcircled{c}$&
  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]
 &
 12, 29, 8, 43, 14, 41, 0, 5, 38, 4, 6, 11, 3, 9, 32]

-&&&\\
+&&\\

 %%%%% n= 7
 \hline
 &
 %%%%% 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]
-&&&\\
-
-
+[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] &&\\

 %%%%%n=8
 \hline
 &
 %%%%%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]
-&&&\\
+[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}
 \hline
 \end{tabular}
 \end{scriptsize}


@@ -219,7 +186,7 @@ achieve to pass the NIST battery of tests.
 
 
 
 
 
 

-\begin{table} 
+\begin{table*} 

 \renewcommand{\arraystretch}{1.3}
 \begin{center}
 \begin{scriptsize}
 \renewcommand{\arraystretch}{1.3}
 \begin{center}
 \begin{scriptsize}


@@ -250,6 +217,12 @@ Linear Complexity& 0.719 (1.0)& 0.739 (0.99)& 0.759 (0.98)& 0.122 (0.97)& 0.514
 \end{center}
 \caption{NIST SP 800-22 test results ($\mathbb{P}_T$)}
 \label{The passing rate}
 \end{center}
 \caption{NIST SP 800-22 test results ($\mathbb{P}_T$)}
 \label{The passing rate}

-\end{table}
+\end{table*}

 
 
 
 

+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "main"
+%%% ispell-dictionary: "american"
+%%% mode: flyspell
+%%% End: