X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/860ecbe3a673a4ac258e24e6c0284a56e3427b6e..cc4c6aa868df1972b090ffcce36c865b8bf53644:/prng_gpu.tex diff --git a/prng_gpu.tex b/prng_gpu.tex index 34ec700..bc40b2d 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -11,6 +11,8 @@ \usepackage[ruled,vlined]{algorithm2e} \usepackage{listings} \usepackage[standard]{ntheorem} +\usepackage{algorithmic} +\usepackage{slashbox} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -416,7 +418,7 @@ the metric space $(\mathcal{X},d)$. \end{proposition} The chaotic property of $G_f$ has been firstly established for the vectorial -Boolean negation $f(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly +Boolean negation $f_0(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly introduced the notion of asynchronous iteration graph recalled bellow. Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The @@ -473,33 +475,58 @@ Let us finally remark that the vectorial negation satisfies the hypotheses of bo We have proposed in~\cite{bgw09:ip} a new family of generators that receives two PRNGs as inputs. These two generators are mixed with chaotic iterations, -leading thus to a new PRNG that improves the statistical properties of each -generator taken alone. Furthermore, our generator -possesses various chaos properties that none of the generators used as input +leading thus to a new PRNG that +\begin{color}{red} +should improves the statistical properties of each +generator taken alone. +Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input present. + \begin{algorithm}[h!] \begin{small} \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)} \KwOut{a configuration $x$ ($n$ bits)} $x\leftarrow x^0$\; -$k\leftarrow b + \textit{XORshift}(b)$\; +$k\leftarrow b + PRNG_1(b)$\; \For{$i=0,\dots,k$} { -$s\leftarrow{\textit{XORshift}(n)}$\; +$s\leftarrow{PRNG_2(n)}$\; $x\leftarrow{F_f(s,x)}$\; } return $x$\; \end{small} -\caption{PRNG with chaotic functions} +\caption{An arbitrary round of $Old~ CI~ PRNG_f(PRNG_1,PRNG_2)$} \label{CI Algorithm} \end{algorithm} +This generator is synthesized in Algorithm~\ref{CI Algorithm}. +It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques}; +an integer $b$, ensuring that the number of executed iterations +between two outputs is at least $b$ +and at most $2b+1$; and an initial configuration $x^0$. +It returns the new generated configuration $x$. Internally, it embeds two +inputted generators $PRNG_i(k), i=1,2$, + which must return integers +uniformly distributed +into $\llbracket 1 ; k \rrbracket$. +For instance, these PRNGs can be the \textit{XORshift}~\cite{Marsaglia2003}, +being a category of very fast PRNGs designed by George Marsaglia +that repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number +with a bit shifted version of it. Such a PRNG, which has a period of +$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. +This XORshift, or any other reasonable PRNG, is used +in our own generator to compute both the number of iterations between two +outputs (provided by $PRNG_1$) and the strategy elements ($PRNG_2$). + +%This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones. + + \begin{algorithm}[h!] \begin{small} \KwIn{the internal configuration $z$ (a 32-bit word)} @@ -515,31 +542,95 @@ return $y$\; \end{algorithm} +\subsection{A ``New CI PRNG''} + +In order to make the Old CI PRNG usable in practice, we have proposed +an adapted version of the chaotic iteration based generator in~\cite{bg10:ip}. +In this ``New CI PRNG'', we prevent from changing twice a given +bit between two outputs. +This new generator is designed by the following process. + +First of all, some chaotic iterations have to be done to generate a sequence +$\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ +of Boolean vectors, which are the successive states of the iterated system. +Some of these vectors will be randomly extracted and our pseudo-random bit +flow will be constituted by their components. Such chaotic iterations are +realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean +vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in +\llbracket 1, 32 \rrbracket^\mathds{N}$ is +an \emph{irregular decimation} of $PRNG_2$ sequence, as described in +Algorithm~\ref{Chaotic iteration1}. + +Then, at each iteration, only the $S^n$-th component of state $x^n$ is +updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$. +Such a procedure is equivalent to achieve chaotic iterations with +the Boolean vectorial negation $f_0$ and some well-chosen strategies. +Finally, some $x^n$ are selected +by a sequence $m^n$ as the pseudo-random bit sequence of our generator. +$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers. + +The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}. +The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input +PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}. +This function is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$ +(the reader is referred to~\cite{bg10:ip} for more information). +\begin{equation} +\label{Formula} +m^n = g(y^n)= +\left\{ +\begin{array}{l} +0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\ +1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\ +2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\ +\vdots~~~~~ ~~\vdots~~~ ~~~~\\ +N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ +\end{array} +\right. +\end{equation} - -This generator is synthesized in Algorithm~\ref{CI Algorithm}. -It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques}; -an integer $b$, ensuring that the number of executed iterations is at least $b$ -and at most $2b+1$; and an initial configuration $x^0$. -It returns the new generated configuration $x$. Internally, it embeds two -\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that return integers -uniformly distributed -into $\llbracket 1 ; k \rrbracket$. -\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia, -which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number -with a bit shifted version of it. This PRNG, which has a period of -$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used -in our PRNG to compute the strategy length and the strategy elements. - -This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones. +\begin{algorithm} +\textbf{Input:} the internal state $x$ (32 bits)\\ +\textbf{Output:} a state $r$ of 32 bits +\begin{algorithmic}[1] +\FOR{$i=0,\dots,N$} +{ +\STATE$d_i\leftarrow{0}$\; +} +\ENDFOR +\STATE$a\leftarrow{PRNG_1()}$\; +\STATE$m\leftarrow{g(a)}$\; +\STATE$k\leftarrow{m}$\; +\WHILE{$i=0,\dots,k$} + +\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\; +\STATE$S\leftarrow{b}$\; + \IF{$d_S=0$} + { +\STATE $x_S\leftarrow{ \overline{x_S}}$\; +\STATE $d_S\leftarrow{1}$\; + + } + \ELSIF{$d_S=1$} + { +\STATE $k\leftarrow{ k+1}$\; + }\ENDIF +\ENDWHILE\\ +\STATE $r\leftarrow{x}$\; +\STATE return $r$\; +\medskip +\caption{An arbitrary round of the new CI generator} +\label{Chaotic iteration1} +\end{algorithmic} +\end{algorithm} +\end{color} \subsection{Improving the Speed of the Former Generator} -Instead of updating only one cell at each iteration, we can try to choose a -subset of components and to update them together. Such an attempt leads -to a kind of merger of the two sequences used in Algorithm -\ref{CI Algorithm}. When the updating function is the vectorial negation, +Instead of updating only one cell at each iteration,\begin{color}{red} we now propose to choose a +subset of components and to update them together, for speed improvements. Such a proposition leads\end{color} +to a kind of merger of the two sequences used in Algorithms +\ref{CI Algorithm} and \ref{Chaotic iteration1}. When the updating function is the vectorial negation, this algorithm can be rewritten as follows: \begin{equation} @@ -549,7 +640,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, \end{array} \right. -\label{equation Oplus} +\label{equation Oplus0} \end{equation} where $\oplus$ is for the bitwise exclusive or between two integers. This rewriting can be understood as follows. The $n-$th term $S^n$ of the @@ -559,7 +650,7 @@ as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th component of this state (a binary digit) changes if and only if the $k-$th digit in the binary decomposition of $S^n$ is 1. -The single basic component presented in Eq.~\ref{equation Oplus} is of +The single basic component presented in Eq.~\ref{equation Oplus0} is of ordinary use as a good elementary brick in various PRNGs. It corresponds to the following discrete dynamical system in chaotic iterations: @@ -580,9 +671,12 @@ than the ones presented in Definition \ref{Def:chaotic iterations} because, inst we select a subset of components to change. -Obviously, replacing Algorithm~\ref{CI Algorithm} by -Equation~\ref{equation Oplus}, which is possible when the iteration function is -the vectorial negation, leads to a speed improvement. However, proofs +Obviously, replacing the previous CI PRNG Algorithms by +Equation~\ref{equation Oplus0}, which is possible when the iteration function is +the vectorial negation, leads to a speed improvement +(the resulting generator will be referred as ``Xor CI PRNG'' +in what follows). +However, proofs of chaos obtained in~\cite{bg10:ij} have been established only for chaotic iterations of the form presented in Definition \ref{Def:chaotic iterations}. The question is now to determine whether the @@ -835,8 +929,189 @@ have $d((S,E),(\tilde S,E))<\epsilon$. \end{proof} +\begin{color}{red} +\section{Statistical Improvements Using Chaotic Iterations} + +\label{The generation of pseudo-random sequence} + + +Let us now explain why we are reasonable grounds to believe that chaos +can improve statistical properties. +We will show in this section that, when mixing defective PRNGs with +chaotic iterations, the result presents better statistical properties +(this section summarizes the work of~\cite{bfg12a:ip}). + +\subsection{Details of some Existing Generators} + +The list of defective PRNGs we will use +as inputs for the statistical tests to come is introduced here. + +Firstly, the simple linear congruency generator (LCGs) will be used. +It is defined by the following recurrence: +\begin{equation} +x^n = (ax^{n-1} + c)~mod~m +\label{LCG} +\end{equation} +where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than +$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) +combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}. + +Secondly, the multiple recursive generators (MRGs) will be used too, which +are based on a linear recurrence of order +$k$, modulo $m$~\cite{LEcuyerS07}: +\begin{equation} +x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m +\label{MRG} +\end{equation} +Combination of two MRGs (referred as 2MRGs) is also used in these experimentations. + +Generators based on linear recurrences with carry will be regarded too. +This family of generators includes the add-with-carry (AWC) generator, based on the recurrence: +\begin{equation} +\label{AWC} +\begin{array}{l} +x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\ +c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation} +the SWB generator, having the recurrence: +\begin{equation} +\label{SWB} +\begin{array}{l} +x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\ +c^n=\left\{ +\begin{array}{l} +1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\ +0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation} +and the SWC generator designed by R. Couture, which is based on the following recurrence: +\begin{equation} +\label{SWC} +\begin{array}{l} +x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\ +c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation} + +Then the generalized feedback shift register (GFSR) generator has been implemented, that is: +\begin{equation} +x^n = x^{n-r} \oplus x^{n-k} +\label{GFSR} +\end{equation} + + +Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is: + +\begin{equation} +\label{INV} +\begin{array}{l} +x^n=\left\{ +\begin{array}{ll} +(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\ +a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation} + + + + + +\subsection{Statistical tests} +\label{Security analysis} + +Three batteries of tests are reputed and usually used +to evaluate the statistical properties of newly designed pseudorandom +number generators. These batteries are named DieHard~\cite{Marsaglia1996}, +the NIST suite~\cite{ANDREW2008}, and the most stringent one called +TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries. + + + +\label{Results and discussion} +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI} +\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|} + \hline\hline +Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline +\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline +NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline +DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline +\end{tabular} +\end{table*} + +Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the +results on the two firsts batteries recalled above, indicating that all the PRNGs presented +in the previous section +cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot +fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic +iterations can solve this issue. +%More precisely, to +%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}: +%\begin{enumerate} +% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category. +% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process. +% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$ +%\begin{equation} +%\begin{array}{l} +%\left\{ +%\begin{array}{l} +%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\ +%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array} +%\end{equation} +%$m$ is called the \emph{functional power}. +%\end{enumerate} +% +The obtained results are reproduced in Table +\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}. +The scores written in boldface indicate that all the tests have been passed successfully, whereas an +asterisk ``*'' means that the considered passing rate has been improved. +The improvements are obvious for both the ``Old CI'' and ``New CI'' generators. +Concerning the ``Xor CI PRNG'', the speed improvement makes that statistical +results are not as good as for the two other versions of these CIPRNGs. + + +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI} +\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|} + \hline +Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline +\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline +Old CIPRNG\\ \hline \hline +NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline +DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline +New CIPRNG\\ \hline \hline +NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline +DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline +Xor CIPRNG\\ \hline\hline +NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline +DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline +\end{tabular} +\end{table*} + + +We have then investigate in~\cite{bfg12a:ip} if it is possible to improve +the statistical behavior of the Xor CI version by combining more than one +$\oplus$ operation. Results are summarized in~\ref{threshold}, showing +that rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained +using chaotic iterations on defective generators. + +\begin{table*} +\renewcommand{\arraystretch}{1.3} +\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries} +\label{threshold} +\centering + \begin{tabular}{|l||c|c|c|c|c|c|c|c|} + \hline +Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline +Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline +\end{tabular} +\end{table*} + +Next subsection gives a concrete implementation of this Xor CI PRNG, which will +new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does not +raise ambiguity. +\end{color} -\section{Efficient PRNG based on Chaotic Iterations} +\subsection{Efficient PRNG based on Chaotic Iterations} \label{sec:efficient PRNG} Based on the proof presented in the previous section, it is now possible to