X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/d509acd9cc478541bd9cf2f33686e60f3b77cce5..dc38289efffc51c12872f9b0be76aefc25b32a05:/prng.tex?ds=inline diff --git a/prng.tex b/prng.tex index 8acfce7..2d1d7f2 100644 --- a/prng.tex +++ b/prng.tex @@ -6,7 +6,7 @@ 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 +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. @@ -46,7 +46,7 @@ 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. +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. @@ -61,7 +61,7 @@ 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 +In this table, let us consider, for instance, the function $\textcircled{a}$ from $\Bool^4$ to $\Bool^4$ defined by the following images : $[13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]$. @@ -71,7 +71,7 @@ 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 $10^{-6}$. +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}). @@ -110,23 +110,25 @@ $\textcircled{c}$& %%%%% n= 7 \hline & -[111, 124, 93, 120, 122, 90, 113, 88, 115, 126, 125, 84, 123, 98, +[111, 124, 93, 120, 122, 114, 89, 121, 87, 126, 125, 84, 123, 82, &&\\ -&112, 96, 109, 106, 77, 110, 99, 74, 104, 72, 71, 100, 117, 116, +&112, 80, 79, 106, 105, 110, 75, 107, 73, 108, 119, 100, 117, 116, &&\\ -&103, 102, 65, 97, 31, 86, 95, 28, 27, 91, 121, 92, 119, 118, 69, +&103, 102, 101, 97, 31, 86, 95, 94, 83, 26, 88, 24, 71, 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, +&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, 94, 61, 24, 26, 18, 17, 25, 19, 30, 85, +&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, &&\\ -&22, 83, 2, 16, 80, 13, 78, 9, 14, 3, 11, 73, 12, 23, 4, 21, 20, +&67, 66, 5, 1] &&\\ -&67, 66, 5, 1] &&\\ + + %%%%%n=8 \hline & @@ -170,7 +172,7 @@ $\textcircled{e}$&151, 149, 19, 210, 144, 152, 141, 206, 13, 12, 171, 10, 201, 1 \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*} @@ -178,42 +180,76 @@ $\textcircled{e}$&151, 149, 19, 210, 144, 152, 141, 206, 13, 12, 171, 10, 201, 1 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{scriptsize} +\begin{tiny} \setlength{\tabcolsep}{2pt} +\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|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|} + \hline +Test +&$\textcircled{a}$& $\textcircled{b}$ & $\textcircled{c}$ & $\textcircled{d}$ & $\textcircled{e}$ \\ \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{scriptsize} + +\end{tiny} \end{center} \caption{NIST SP 800-22 test results ($\mathbb{P}_T$)} \label{The passing rate}