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
+a random choice of which edge to follow, and crosses 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.
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.
+only requires the graph $\Gamma_{\{b\}}(f)$ to be strongly connected.
Since the $\chi_{\textit{16HamG}}$ algorithm
only adds probability constraints on existing edges,
it preserves this property.
has the smallest practical mixing time.
Such functions are
given in Table~\ref{table:nc}.
-In this table, let us consider for instance
+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 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 $10^{-6}$.
+where the deviation to the standard distribution is inferior than $10^{-6}$.
%(resp. the theoretical upper bound of stopping time as described in Sect.~\ref{sec:hypercube}).
\end{tabular}
\end{scriptsize}
\end{center}
-\caption{Functions with DSCC Matrix and smallest MT\label{table:nc}}
+\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
+and the generator is unsuitable.
+
+Table~\ref{The passing rate} shows $\mathbb{P}_T$ of sequences based
+on $\chi_{\textit{16HamG}}$ using different functions, namely
+$\textcircled{a}$,\ldots, $\textcircled{e}$.
+In this algorithm implementation,
+the embedded PRNG \textit{Random} is the default Python PRNG, \textit{i.e.},
+the Mersenne Twister algorithm~\cite{matsumoto1998mersenne}.
+Implementations for $\mathsf{N}=4, \dots, 8$ of this algorithm is evaluated
+through the NIST test suite and results are given in columns
+$\textit{MT}_4$, \ldots, $\textit{MT}_8$.
+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.
+We first 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.
+It can be noticed that adding chaos properties for Mersenne Twister
+algorithm does not reduce its security against this statistical tests.
\begin{table*}
-\renewcommand{\arraystretch}{1.3}
+\renewcommand{\arraystretch}{1.1}
\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|}
+\begin{tabular}{|l|r|r|r|r|r|}
\hline
Test & $\textit{MT}_4$ & $\textit{MT}_5$& $\textit{MT}_6$& $\textit{MT}_7$& $\textit{MT}_8$
+ \\ \hline
+Frequency (Monobit)& 0.924 (1.0)& 0.678 (0.98)& 0.102 (0.97)& 0.213 (0.98)& 0.719 (0.99) \\ \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)\\ \hline
+Cumulative Sums (Cusum) *& 0.668 (1.0)& 0.568 (0.99)& 0.881 (0.98)& 0.529 (0.98)& 0.657 (0.995)\\ \hline
+Runs& 0.494 (0.99)& 0.595 (0.97)& 0.071 (0.97)& 0.017 (1.0)& 0.834 (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)\\ \hline
+Binary Matrix Rank& 0.275 (0.98)& 0.494 (0.99)& 0.719 (1.0)& 0.334 (0.98)& 0.637 (0.99)\\ \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)\\ \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)\\ \hline
+Overlapping Template Matching& 0.595 (0.99)& 0.759 (1.0)& 0.637 (1.0)& 0.554 (0.99)& 0.236 (1.0)\\ \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)\\ \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)\\ \hline
+Random Excursions *& 0.275 (1.0)& 0.495 (0.975)& 0.465 (0.979)& 0.452 (0.991)& 0.260 (0.989)\\ \hline
+Random Excursions Variant *& 0.382 (0.995)& 0.400 (0.994)& 0.417 (0.984)& 0.456 (0.991)& 0.389 (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)\\ \hline
+Linear Complexity& 0.494 (0.99)& 0.514 (0.98)& 0.145 (1.0)& 0.657 (0.98)& 0.145 (0.99)\\ \hline
+\end{tabular}
+
+\begin{tabular}{|l|r|r|r|r|r|}
+ \hline
+Test
&$\textcircled{a}$& $\textcircled{b}$ & $\textcircled{c}$ & $\textcircled{d}$ & $\textcircled{e}$ \\ \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
+Frequency (Monobit)&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& 0.275 (1.0)& 0.534 (0.98)& 0.066 (1.0)& 0.719 (1.0)& 0.366 (1.0)\\ \hline
+Cumulative Sums (Cusum) *& 0.695 (1.0)& 0.540 (1.0)& 0.514 (0.985)& 0.773 (0.995)& 0.506 (0.99)\\ \hline
+Runs& 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& 0.851 (1.0)& 0.595 (0.99)& 0.419 (0.98)& 0.616 (0.98)& 0.897 (1.0)\\ \hline
+Binary Matrix Rank& 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)& 0.867 (1.0)& 0.514 (1.0)& 0.145 (1.0)& 0.224 (0.99)& 0.304 (1.0)\\ \hline
+Non-overlapping Template Matching*& 0.542 (0.990)& 0.512 (0.989)& 0.505 (0.990)& 0.494 (0.989)& 0.493 (0.991)\\ \hline
+Overlapping Template Matching& 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"& 0.383 (0.99)& 0.991 (0.98)& 0.851 (1.0)& 0.595 (0.98)& 0.514 (1.0)\\ \hline
+Approximate Entropy (m=10)& 0.935 (1.0)& 0.719 (1.0)& 0.883 (1.0)& 0.719 (0.97)& 0.366 (0.99)\\ \hline
+Random Excursions *& 0.396 (0.991)& 0.217 (0.989)& 0.445 (0.975)& 0.743 (0.993)& 0.380 (0.990)\\ \hline
+Random Excursions Variant *& 0.486 (0.997)& 0.373 (0.981)& 0.415 (0.994)& 0.424 (0.991)& 0.380 (0.991)\\ \hline
+Serial* (m=10)&0.350 (1.0)& 0.678 (0.995)& 0.287 (0.995)& 0.740 (0.99)& 0.301 (0.98)\\ \hline
+Linear Complexity& 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 ($\mathbb{P}_T$)}
