X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/26aa4e0cf2939580eae07e8bbad140d527f47234..850c033d45e9af70be22cb2e0c76a9de99d23c17:/prng_gpu.tex?ds=inline diff --git a/prng_gpu.tex b/prng_gpu.tex index 0279f03..966dbaa 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1,4 +1,5 @@ -\documentclass{article} +%\documentclass{article} +\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{fullpage} @@ -7,9 +8,11 @@ \usepackage{amscd} \usepackage{moreverb} \usepackage{commath} -\usepackage{algorithm2e} +\usepackage[ruled,vlined]{algorithm2e} \usepackage{listings} \usepackage[standard]{ntheorem} +\usepackage{algorithmic} +\usepackage{slashbox} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -38,27 +41,35 @@ \begin{document} \author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe -Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}} +Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}} -\maketitle +\IEEEcompsoctitleabstractindextext{ \begin{abstract} In this paper we present a new pseudorandom number generator (PRNG) on graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations. It is firstly proven to be chaotic according to the Devaney's formulation. We thus propose an efficient implementation for GPU that successfully passes the {\it BigCrush} tests, deemed to be the hardest battery of tests in TestU01. Experiments show that this PRNG can generate -about 20 billions of random numbers per second on Tesla C1060 and NVidia GTX280 +about 20 billion of random numbers per second on Tesla C1060 and NVidia GTX280 cards. -It is finally established that, under reasonable assumptions, the proposed PRNG can be cryptographically +It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically secure. +A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is finally proposed. \end{abstract} +} + +\maketitle + +\IEEEdisplaynotcompsoctitleabstractindextext +\IEEEpeerreviewmaketitle + \section{Introduction} -Randomness is of importance in many fields as scientific simulations or cryptography. +Randomness is of importance in many fields such as scientific simulations or cryptography. ``Random numbers'' can mainly be generated either by a deterministic and reproducible algorithm called a pseudorandom number generator (PRNG), or by a physical non-deterministic process having all the characteristics of a random noise, called a truly random number @@ -66,21 +77,21 @@ generator (TRNG). In this paper, we focus on reproducible generators, useful for instance in Monte-Carlo based simulators or in several cryptographic schemes. These domains need PRNGs that are statistically irreproachable. -On some fields as in numerical simulations, speed is a strong requirement +In some fields such as in numerical simulations, speed is a strong requirement that is usually attained by using parallel architectures. In that case, -a recurrent problem is that a deflate of the statistical qualities is often +a recurrent problem is that a deflation of the statistical qualities is often reported, when the parallelization of a good PRNG is realized. This is why ad-hoc PRNGs for each possible architecture must be found to achieve both speed and randomness. On the other side, speed is not the main requirement in cryptography: the great -need is to define \emph{secure} generators being able to withstand malicious +need is to define \emph{secure} generators able to withstand malicious attacks. Roughly speaking, an attacker should not be able in practice to make the distinction between numbers obtained with the secure generator and a true random sequence. -Finally, a small part of the community working in this domain focus on a +Finally, a small part of the community working in this domain focuses on a third requirement, that is to define chaotic generators. The main idea is to take benefits from a chaotic dynamical system to obtain a -generator that is unpredictable, disordered, sensible to its seed, or in other words chaotic. +generator that is unpredictable, disordered, sensible to its seed, or in other word chaotic. Their desire is to map a given chaotic dynamics into a sequence that seems random and unassailable due to chaos. However, the chaotic maps used as a pattern are defined in the real line @@ -94,7 +105,7 @@ This is why the use of chaos for PRNG still remains marginal and disputable. The authors' opinion is that topological properties of disorder, as they are properly defined in the mathematical theory of chaos, can reinforce the quality of a PRNG. But they are not substitutable for security or statistical perfection. -Indeed, to the authors' point of view, such properties can be useful in the two following situations. On the +Indeed, to the authors' mind, such properties can be useful in the two following situations. On the one hand, a post-treatment based on a chaotic dynamical system can be applied to a PRNG statistically deflective, in order to improve its statistical properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}. @@ -109,7 +120,7 @@ Let us finish this paragraph by noticing that, in this paper, statistical perfection refers to the ability to pass the whole {\it BigCrush} battery of tests, which is widely considered as the most stringent statistical evaluation of a sequence claimed as random. -This battery can be found into the well-known TestU01 package~\cite{LEcuyerS07}. +This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}. Chaos, for its part, refers to the well-established definition of a chaotic dynamical system proposed by Devaney~\cite{Devaney}. @@ -130,28 +141,31 @@ applications. Therefore, it is important to be able to generate pseudorandom numbers inside a GPU when a scientific application runs in it. This remark motivates our proposal of a chaotic and statistically perfect PRNG for GPU. Such device -allows us to generated almost 20 billions of pseudorandom numbers per second. -Last, but not least, we show that the proposed post-treatment preserves the +allows us to generate almost 20 billion of pseudorandom numbers per second. +Furthermore, we show that the proposed post-treatment preserves the cryptographical security of the inputted PRNG, when this last has such a property. +Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric +key encryption protocol by using the proposed method. The remainder of this paper is organized as follows. In Section~\ref{section:related works} we review some GPU implementations of PRNGs. Section~\ref{section:BASIC RECALLS} gives some basic recalls on the well-known Devaney's formulation of chaos, and on an iteration process called ``chaotic iterations'' on which the post-treatment is based. -Proofs of chaos are given in Section~\ref{sec:pseudorandom}. +The proposed PRNG and its proof of chaos are given in Section~\ref{sec:pseudorandom}. Section~\ref{sec:efficient PRNG} presents an efficient implementation of this chaotic PRNG on a CPU, whereas Section~\ref{sec:efficient PRNG - gpu} describes the GPU implementation. + gpu} describes and evaluates theoretically the GPU implementation. Such generators are experimented in Section~\ref{sec:experiments}. We show in Section~\ref{sec:security analysis} that, if the inputted generator is cryptographically secure, then it is the case too for the generator provided by the post-treatment. Such a proof leads to the proposition of a cryptographically secure and -chaotic generator on GPU based on the famous Blum Blum Shum -in Section~\ref{sec:CSGPU}. +chaotic generator on GPU based on the famous Blum Blum Shub +in Section~\ref{sec:CSGPU}, and to an improvement of the +Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}. This research work ends by a conclusion section, in which the contribution is summarized and intended future work is presented. @@ -161,8 +175,8 @@ summarized and intended future work is presented. \section{Related works on GPU based PRNGs} \label{section:related works} -Numerous research works on defining GPU based PRNGs have yet been proposed in the -literature, so that completeness is impossible. +Numerous research works on defining GPU based PRNGs have already been proposed in the +literature, so that exhaustivity is impossible. This is why authors of this document only give reference to the most significant attempts in this domain, from their subjective point of view. The quantity of pseudorandom numbers generated per second is mentioned here @@ -180,7 +194,7 @@ chaos or cryptography in this document. In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs based on Lagged Fibonacci or Hybrid Taus. They have used these PRNGs for Langevin simulations of biomolecules fully implemented on -GPU. Performance of the GPU versions are far better than those obtained with a +GPU. Performances of the GPU versions are far better than those obtained with a CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. However the evaluations of the proposed PRNGs are only statistical ones. @@ -192,11 +206,11 @@ the performance of the same PRNGs on different architectures are compared. FPGA appears as the fastest and the most efficient architecture, providing the fastest number of generated pseudorandom numbers per joule. -However, we can notice that authors can ``only'' generate between 11 and 16GSamples/s +However, we notice that authors can ``only'' generate between 11 and 16GSamples/s with a GTX 280 GPU, which should be compared with the results presented in this document. We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only -able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one. +able to pass the {\it Crush} battery, which is far easier than the {\it Big Crush} one. Lastly, Cuda has developed a library for the generation of pseudorandom numbers called Curand~\cite{curand11}. Several PRNGs are implemented, among @@ -206,13 +220,16 @@ their fastest version provides 15GSamples/s on the new Fermi C2050 card. But their PRNGs cannot pass the whole TestU01 battery (only one test is failed). \newline \newline -We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded. +We can finally remark that, to the best of our knowledge, no GPU implementation has been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered. \section{Basic Recalls} \label{section:BASIC RECALLS} This section is devoted to basic definitions and terminologies in the fields of -topological chaos and chaotic iterations. +topological chaos and chaotic iterations. We assume the reader is familiar +with basic notions on topology (see for instance~\cite{Devaney}). + + \subsection{Devaney's Chaotic Dynamical Systems} In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ @@ -225,7 +242,7 @@ Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : \mathcal{X} \rightarrow \mathcal{X}$. \begin{definition} -$f$ is said to be \emph{topologically transitive} if, for any pair of open sets +The function $f$ is said to be \emph{topologically transitive} if, for any pair of open sets $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq \varnothing$. \end{definition} @@ -244,7 +261,7 @@ necessarily the same period). \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}] -$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and +The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and topologically transitive. \end{definition} @@ -252,12 +269,12 @@ The chaos property is strongly linked to the notion of ``sensitivity'', defined on a metric space $(\mathcal{X},d)$ by: \begin{definition} -\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions} +\label{sensitivity} The function $f$ has \emph{sensitive dependence on initial conditions} if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that $d\left(f^{n}(x), f^{n}(y)\right) >\delta $. -$\delta$ is called the \emph{constant of sensitivity} of $f$. +The constant $\delta$ is called the \emph{constant of sensitivity} of $f$. \end{definition} Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is @@ -316,15 +333,15 @@ Let us now recall how to define a suitable metric space where chaotic iterations are continuous. For further explanations, see, e.g., \cite{guyeux10}. Let $\delta $ be the \emph{discrete Boolean metric}, $\delta -(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function: -\begin{equation} +(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function +$F_{f}: \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} +\longrightarrow \mathds{B}^{\mathsf{N}}$ +\begin{equation*} \begin{array}{lrll} -F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} & -\longrightarrow & \mathds{B}^{\mathsf{N}} \\ -& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta -(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% +& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ f(E)_{k}.\overline{\delta +(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}% \end{array}% -\end{equation}% +\end{equation*}% \noindent where + and . are the Boolean addition and product operations. Consider the phase space: \begin{equation} @@ -380,9 +397,9 @@ their distance should increase too. \item In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a -while. Indeed, the two dynamical systems start with the same initial condition, -use the same update function, and as strategies are the same for a while, then -components that are updated are the same too. +while. Indeed, both dynamical systems start with the same initial condition, +use the same update function, and as strategies are the same for a while, furthermore +updated components are the same as well. \end{itemize} The distance presented above follows these recommendations. Indeed, if the floor value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ @@ -391,7 +408,7 @@ measure of the differences between strategies $S$ and $\check{S}$. More precisely, this floating part is less than $10^{-k}$ if and only if the first $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the $k^{th}$ terms of the two strategies are different. -The impact of this choice for a distance will be investigate at the end of the document. +The impact of this choice for a distance will be investigated at the end of the document. Finally, it has been established in \cite{guyeux10} that, @@ -401,7 +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 @@ -414,8 +431,7 @@ The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a strategy $s$ such that the parallel iteration of $G_f$ from the initial point $(s,x)$ reaches the point $x'$. - -We have finally proven in \cite{bcgr11:ip} that, +We have then proven in \cite{bcgr11:ip} that, \begin{theorem} @@ -424,14 +440,33 @@ Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic (accord if and only if $\Gamma(f)$ is strongly connected. \end{theorem} -This result of chaos has lead us to study the possibility to build a +Finally, we have established in \cite{bcgr11:ip} that, +\begin{theorem} + Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its + iteration graph, $\check{M}$ its adjacency + matrix and $M$ + a $n\times n$ matrix defined by + $ + M_{ij} = \frac{1}{n}\check{M}_{ij}$ %\textrm{ + if $i \neq j$ and + $M_{ii} = 1 - \frac{1}{n} \sum\limits_{j=1, j\neq i}^n \check{M}_{ij}$ otherwise. + + If $\Gamma(f)$ is strongly connected, then + the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows + a law that tends to the uniform distribution + if and only if $M$ is a double stochastic matrix. +\end{theorem} + + +These results of chaos and uniform distribution have led us to study the possibility of building a pseudorandom number generator (PRNG) based on the chaotic iterations. As $G_f$, defined on the domain $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}} -\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N} +\times \mathds{B}^\mathsf{N}$, is built from Boolean networks $f : \mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$ -during implementations (due to the discrete nature of $f$). It is as if +during implementations (due to the discrete nature of $f$). Indeed, it is as if $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG). +Let us finally remark that the vectorial negation satisfies the hypotheses of both theorems above. \section{Application to Pseudorandomness} \label{sec:pseudorandom} @@ -440,30 +475,60 @@ $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracke 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{scriptsize} +\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{scriptsize} -\caption{PRNG with chaotic functions} +\end{small} +\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)} \KwOut{$y$ (a 32-bit word)} $z\leftarrow{z\oplus{(z\ll13)}}$\; @@ -471,49 +536,101 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\; $z\leftarrow{z\oplus{(z\ll5)}}$\; $y\leftarrow{z}$\; return $y$\; -\medskip +\end{small} \caption{An arbitrary round of \textit{XORshift} algorithm} \label{XORshift} \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 returns 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. - - -We have proven in \cite{bcgr11:ip} that, -\begin{theorem} - Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its - iteration graph, $\check{M}$ its adjacency - matrix and $M$ a $n\times n$ matrix defined as in the previous lemma. - If $\Gamma(f)$ is strongly connected, then - the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows - a law that tends to the uniform distribution - if and only if $M$ is a double stochastic matrix. -\end{theorem} - -This former generator as successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}. +\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} @@ -523,17 +640,17 @@ 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 rewritten can be understood as follows. The $n-$th term $S^n$ of the +This rewriting can be understood as follows. The $n-$th term $S^n$ of the sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents the list of cells to update in the state $x^n$ of the system (represented 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: @@ -550,14 +667,16 @@ where $f$ is the vectorial negation and $\forall n \in \mathds{N}$, $\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that $k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary decomposition of $S^n$ is 1. Such chaotic iterations are more general -than the ones presented in Definition \ref{Def:chaotic iterations} for -the fact that, instead of updating only one term at each iteration, +than the ones presented in Definition \ref{Def:chaotic iterations} because, instead of updating only one term at each iteration, we select a subset of components to change. -Obviously, replacing Algorithm~\ref{CI Algorithm} by -Equation~\ref{equation Oplus}, 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 @@ -567,11 +686,11 @@ faster, does not deflate their topological chaos properties. \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations} \label{deuxième def} Let us consider the discrete dynamical systems in chaotic iterations having -the general form: +the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in +\llbracket1;\mathsf{N}\rrbracket $, \begin{equation} -\forall n\in \mathds{N}^{\ast }, \forall i\in -\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{ + x_i^n=\left\{ \begin{array}{ll} x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n. @@ -596,14 +715,13 @@ Let us introduce the following function: where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$. Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function: -\begin{equation} -\begin{array}{lrll} -F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} & -\longrightarrow & \mathds{B}^{\mathsf{N}} \\ -& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi -(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% +$F_{f}: \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} +\longrightarrow \mathds{B}^{\mathsf{N}}$ +\begin{equation*} +\begin{array}{rll} + (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}% \end{array}% -\end{equation}% +\end{equation*}% where + and . are the Boolean addition and product operations, and $\overline{x}$ is the negation of the Boolean $x$. Consider the phase space: @@ -613,7 +731,7 @@ Consider the phase space: \end{equation} \noindent and the map defined on $\mathcal{X}$: \begin{equation} -G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf} +G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai viré ce label qui existe déjà avant... \end{equation} \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma (S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in @@ -630,7 +748,7 @@ X^{k+1}=G_{f}(X^k).% \right. \end{equation}% -Another time, a shift function appears as a component of these general chaotic +Once more, a shift function appears as a component of these general chaotic iterations. To study the Devaney's chaos property, a distance between two points @@ -640,17 +758,21 @@ Let us introduce: d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}), \label{nouveau d} \end{equation} -\noindent where -\begin{equation} -\left\{ -\begin{array}{lll} -\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}% -}\delta (E_{k},\check{E}_{k})}\textrm{ is another time the Hamming distance}, \\ -\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}% -\sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.% -\end{array}% -\right. -\end{equation} +\noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}% + }\delta (E_{k},\check{E}_{k})}$ is once more the Hamming distance, and +$ \displaystyle{d_{s}(S,\check{S})} = \displaystyle{\dfrac{9}{\mathsf{N}}% + \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$, +%%RAPH : ici, j'ai supprimé tous les sauts à la ligne +%% \begin{equation} +%% \left\{ +%% \begin{array}{lll} +%% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}% +%% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\ +%% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}% +%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.% +%% \end{array}% +%% \right. +%% \end{equation} where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$. @@ -661,7 +783,7 @@ The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$. \begin{proof} $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance -too, thus $d$ will be a distance as sum of two distances. +too, thus $d$, as being the sum of two distances, will also be a distance. \begin{itemize} \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then @@ -678,7 +800,7 @@ inequality is obtained. Before being able to study the topological behavior of the general -chaotic iterations, we must firstly establish that: +chaotic iterations, we must first establish that: \begin{proposition} For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on @@ -714,7 +836,7 @@ so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two poin G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to 0. Let $\varepsilon >0$. \medskip \begin{itemize} -\item If $\varepsilon \geqslant 1$, we see that distance +\item If $\varepsilon \geqslant 1$, we see that the distance between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state). \medskip @@ -729,14 +851,16 @@ thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal. \noindent As a consequence, the $k+1$ first entries of the strategies of $% G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $% -10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline +10^{-(k+1)}\leqslant \varepsilon $. + In conclusion, -$$ -\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}% -,\forall n\geqslant N_{0}, - d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) +%%RAPH : ici j'ai rajouté une ligne +$ +\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N} +,$ $\forall n\geqslant N_{0},$ +$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) \leqslant \varepsilon . -$$ +$ $G_{f}$ is consequently continuous. \end{proof} @@ -776,7 +900,7 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties claimed in the lemma. \end{proof} -We can now prove the Theorem~\ref{t:chaos des general}... +We can now prove the Theorem~\ref{t:chaos des general}. \begin{proof}[Theorem~\ref{t:chaos des general}] Firstly, strong transitivity implies transitivity. @@ -794,8 +918,10 @@ and $t_2\in\mathds{N}$ such that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$. Consider the strategy $\tilde S$ that alternates the first $t_1$ terms -of $S$ and the first $t_2$ terms of $S'$: $$\tilde -S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It +of $S$ and the first $t_2$ terms of $S'$: +%%RAPH : j'ai coupé la ligne en 2 +$$\tilde +S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after $t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic point. Since $\tilde S_t=S_t$ for $t>$32)} in order to obtain the 32 most significant bits of \texttt{t}. - -In Listing~\ref{algo:seqCIPRNG} a sequential version of the proposed PRNG based on chaotic iterations - is presented. The xor operator is represented by \textasciicircum. -This function uses three classical 64-bits PRNGs, namely the \texttt{xorshift}, the -\texttt{xor128}, and the \texttt{xorwow}~\cite{Marsaglia2003}. In the following, we call them -``xor-like PRNGs''. -As -each xor-like PRNG uses 64-bits whereas our proposed generator works with 32-bits, -we use the command \texttt{(unsigned int)}, that selects the 32 least significant bits of a given integer, and the code -\texttt{(unsigned int)(t3$>>$32)} in order to obtain the 32 most significant bits of \texttt{t}. - -So producing a pseudorandom number needs 6 xor operations -with 6 32-bits numbers that are provided by 3 64-bits PRNGs. This version successfully passes the +Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers +that are provided by 3 64-bits PRNGs. This version successfully passes the stringent BigCrush battery of tests~\cite{LEcuyerS07}. \section{Efficient PRNGs based on Chaotic Iterations on GPU} \label{sec:efficient PRNG gpu} -In order to take benefits from the computing power of GPU, a program needs to have -independent blocks of threads that can be computed simultaneously. In general, -the larger the number of threads is, the more local memory is used, and the less -branching instructions are used (if, while, ...), the better the performances on GPU is. -Obviously, having these requirements in mind, it is possible to build a program similar to -the one presented in Algorithm \ref{algo:seqCIPRNG}, which computes pseudorandom numbers -on GPU. -To do so, we must firstly recall that in - the CUDA~\cite{Nvid10} environment, threads have a local -identifier called \texttt{ThreadIdx}, which is relative to the block containing them. +In order to take benefits from the computing power of GPU, a program +needs to have independent blocks of threads that can be computed +simultaneously. In general, the larger the number of threads is, the +more local memory is used, and the less branching instructions are +used (if, while, ...), the better the performances on GPU is. +Obviously, having these requirements in mind, it is possible to build +a program similar to the one presented in Listing +\ref{algo:seqCIPRNG}, which computes pseudorandom numbers on GPU. To +do so, we must firstly recall that in the CUDA~\cite{Nvid10} +environment, threads have a local identifier called +\texttt{ThreadIdx}, which is relative to the block containing +them. Furthermore, in CUDA, parts of the code that are executed by the GPU, are +called {\it kernels}. \subsection{Naive Version for GPU} It is possible to deduce from the CPU version a quite similar version adapted to GPU. -The simple principle consists to make each thread of the GPU computing the CPU version of our PRNG. +The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG. Of course, the three xor-like PRNGs used in these computations must have different parameters. -In a given thread, these lasts are +In a given thread, these parameters are randomly picked from another PRNGs. The initialization stage is performed by the CPU. To do it, the ISAAC PRNG~\cite{Jenkins96} is used to set all the @@ -916,8 +1230,9 @@ number $x$ that saves the last generated pseudorandom number. Additionally implementation of the xor128, the xorshift, and the xorwow respectively require 4, 5, and 6 unsigned long as internal variables. -\begin{algorithm} +\begin{algorithm} +\begin{small} \KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like PRNGs in global memory\; NumThreads: number of threads\;} @@ -930,14 +1245,16 @@ NumThreads: number of threads\;} } store internal variables in InternalVarXorLikeArray[threadIdx]\; } - +\end{small} \caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations} \label{algo:gpu_kernel} \end{algorithm} + + Algorithm~\ref{algo:gpu_kernel} presents a naive implementation of the proposed PRNG on GPU. Due to the available memory in the GPU and the number of threads -used simultenaously, the number of random numbers that a thread can generate +used simultaneously, the number of random numbers that a thread can generate inside a kernel is limited (\emph{i.e.}, the variable \texttt{n} in algorithm~\ref{algo:gpu_kernel}). For instance, if $100,000$ threads are used and if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}, @@ -948,14 +1265,14 @@ and the pseudorandom numbers generated by our PRNG, is equal to $100,000\ This generator is able to pass the whole BigCrush battery of tests, for all the versions that have been tested depending on their number of threads -(called \texttt{NumThreads} in our algorithm, tested until $10$ millions). +(called \texttt{NumThreads} in our algorithm, tested up to $5$ million). \begin{remark} -The proposed algorithm has the advantage to manipulate independent +The proposed algorithm has the advantage of manipulating independent PRNGs, so this version is easily adaptable on a cluster of computers too. The only thing to ensure is to use a single ISAAC PRNG. To achieve this requirement, a simple solution consists in using a master node for the initialization. This master node computes the initial parameters -for all the differents nodes involves in the computation. +for all the different nodes involved in the computation. \end{remark} \subsection{Improved Version for GPU} @@ -965,46 +1282,48 @@ is possible to use this feature in order to simplify the previous algorithm, i.e., to use less than 3 xor-like PRNGs. The solution consists in computing only one xor-like PRNG by thread, saving it into the shared memory, and then to use the results of some other threads in the same block of threads. In order to define which -thread uses the result of which other one, we can use a permutation array that -contains the indexes of all threads and for which a permutation has been +thread uses the result of which other one, we can use a combination array that +contains the indexes of all threads and for which a combination has been performed. -In Algorithm~\ref{algo:gpu_kernel2}, two permutations arrays are used. -The variable \texttt{offset} is computed using the value of -\texttt{permutation\_size}. Then we can compute \texttt{o1} and \texttt{o2} -representing the indexes of the other threads whose results are used -by the current one. In this algorithm, we consider that a 64-bits xor-like -PRNG has been chosen, and so its two 32-bits parts are used. +In Algorithm~\ref{algo:gpu_kernel2}, two combination arrays are used. The +variable \texttt{offset} is computed using the value of +\texttt{combination\_size}. Then we can compute \texttt{o1} and \texttt{o2} +representing the indexes of the other threads whose results are used by the +current one. In this algorithm, we consider that a 32-bits xor-like PRNG has +been chosen. In practice, we use the xor128 proposed in~\cite{Marsaglia2003} in +which unsigned longs (64 bits) have been replaced by unsigned integers (32 +bits). -This version also can pass the whole {\it BigCrush} battery of tests. +This version can also pass the whole {\it BigCrush} battery of tests. \begin{algorithm} - +\begin{small} \KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs in global memory\; NumThreads: Number of threads\; -tab1, tab2: Arrays containing permutations of size permutation\_size\;} +array\_comb1, array\_comb2: Arrays containing combinations of size combination\_size\;} \KwOut{NewNb: array containing random numbers in global memory} \If{threadId is concerned} { retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory and x\; - offset = threadIdx\%permutation\_size\; - o1 = threadIdx-offset+tab1[offset]\; - o2 = threadIdx-offset+tab2[offset]\; + offset = threadIdx\%combination\_size\; + o1 = threadIdx-offset+array\_comb1[offset]\; + o2 = threadIdx-offset+array\_comb2[offset]\; \For{i=1 to n} { t=xor-like()\; - t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\; + t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\; shared\_mem[threadId]=t\; - x = x $\oplus$ t\; + x = x\textasciicircum t\; store the new PRNG in NewNb[NumThreads*threadId+i]\; } store internal variables in InternalVarXorLikeArray[threadId]\; } - -\caption{main kernel for the chaotic iterations based PRNG GPU efficient -version} -\label{algo:gpu_kernel2} +\end{small} +\caption{Main kernel for the chaotic iterations based PRNG GPU efficient +version\label{IR}} +\label{algo:gpu_kernel2} \end{algorithm} \subsection{Theoretical Evaluation of the Improved Version} @@ -1018,7 +1337,7 @@ and two values previously obtained by two other threads). To be certain that we are in the framework of Theorem~\ref{t:chaos des general}, we must guarantee that this dynamical system iterates on the space $\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$. -The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$. +The left term $x$ obviously belongs to $\mathds{B}^ \mathsf{N}$. To prevent from any flaws of chaotic properties, we must check that the right term (the last $t$), corresponding to the strategies, can possibly be equal to any integer of $\llbracket 1, \mathsf{N} \rrbracket$. @@ -1029,8 +1348,7 @@ last $t$ respects the requirement. Furthermore, it is possible to prove by an immediate mathematical induction that, as the initial $x$ is uniformly distributed (it is provided by a cryptographically secure PRNG), the two other stored values shmem[o1] and shmem[o2] are uniformly distributed too, -(this can be stated by an immediate mathematical -induction), and thus the next $x$ is finally uniformly distributed. +(this is the induction hypothesis), and thus the next $x$ is finally uniformly distributed. Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general chaotic iterations presented previously, and for this reason, it satisfies the @@ -1048,7 +1366,7 @@ All the cards have 240 cores. In Figure~\ref{fig:time_xorlike_gpu} we compare the quantity of pseudorandom numbers -generated per second with various xor-like based PRNG. In this figure, the optimized +generated per second with various xor-like based PRNGs. In this figure, the optimized versions use the {\it xor64} described in~\cite{Marsaglia2003}, whereas the naive versions embed the three xor-like PRNGs described in Listing~\ref{algo:seqCIPRNG}. In order to obtain the optimal performances, the storage of pseudorandom numbers @@ -1056,7 +1374,7 @@ into the GPU memory has been removed. This step is time consuming and slows down generation. Moreover this storage is completely useless, in case of applications that consume the pseudorandom numbers directly after generation. We can see that when the number of threads is greater -than approximately 30,000 and lower than 5 millions, the number of pseudorandom numbers generated +than approximately 30,000 and lower than 5 million, the number of pseudorandom numbers generated per second is almost constant. With the naive version, this value ranges from 2.5 to 3GSamples/s. With the optimized version, it is approximately equal to 20GSamples/s. Finally we can remark that both GPU cards are quite similar, but in @@ -1067,7 +1385,7 @@ As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of \begin{figure}[htbp] \begin{center} - \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf} + \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf} \end{center} \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG} \label{fig:time_xorlike_gpu} @@ -1077,17 +1395,16 @@ As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of -In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized -BBS-based PRNG on GPU. On the Tesla C1060 we -obtain approximately 1.8GSample/s and on the GTX 280 about 1.6GSample/s, which is -obviously slower than the xorlike-based PRNG on GPU. However, we will show in the -next sections that -this new PRNG has a strong level of security, which is necessary paid by a speed -reduction. +In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized +BBS-based PRNG on GPU. On the Tesla C1060 we obtain approximately 700MSample/s +and on the GTX 280 about 670MSample/s, which is obviously slower than the +xorlike-based PRNG on GPU. However, we will show in the next sections that this +new PRNG has a strong level of security, which is necessarily paid by a speed +reduction. \begin{figure}[htbp] \begin{center} - \includegraphics[scale=.7]{curve_time_bbs_gpu.pdf} + \includegraphics[width=\columnwidth]{curve_time_bbs_gpu.pdf} \end{center} \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG} \label{fig:time_bbs_gpu} @@ -1095,7 +1412,7 @@ reduction. All these experiments allow us to conclude that it is possible to generate a very large quantity of pseudorandom numbers statistically perfect with the xor-like version. -In a certain extend, it is the case too with the secure BBS-based version, the speed deflation being +To a certain extend, it is also the case with the secure BBS-based version, the speed deflation being explained by the fact that the former version has ``only'' chaotic properties and statistical perfection, whereas the latter is also cryptographically secure, as it is shown in the next sections. @@ -1115,17 +1432,17 @@ In this section the concatenation of two strings $u$ and $v$ is classically denoted by $uv$. In a cryptographic context, a pseudorandom generator is a deterministic algorithm $G$ transforming strings into strings and such that, for any -seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size -$\ell_G(N)$ with $\ell_G(N)>N$. +seed $s$ of length $m$, $G(s)$ (the output of $G$ on the input $s$) has size +$\ell_G(m)$ with $\ell_G(m)>m$. The notion of {\it secure} PRNGs can now be defined as follows. \begin{definition} A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time algorithm $D$, for any positive polynomial $p$, and for all sufficiently -large $k$'s, -$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(N)},$$ +large $m$'s, +$$| \mathrm{Pr}[D(G(U_m))=1]-Pr[D(U_{\ell_G(m)})=1]|< \frac{1}{p(m)},$$ where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the -probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the +probabilities are taken over $U_m$, $U_{\ell_G(m)}$ as well as over the internal coin tosses of $D$. \end{definition} @@ -1134,7 +1451,7 @@ distinguish a perfect uniform random generator from $G$ with a non negligible probability. The interested reader is referred to~\cite[chapter~3]{Goldreich} for more information. Note that it is quite easily possible to change the function $\ell$ into any polynomial -function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}. +function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}. The generation schema developed in (\ref{equation Oplus}) is based on a pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume, @@ -1145,7 +1462,7 @@ strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concate the $S_i$'s). The cryptographic PRNG $X$ defined in (\ref{equation Oplus}) is the algorithm mapping any string of length $2N$ $x_0S_0$ into the string $(x_0\oplus S_0 \oplus S_1)(x_0\oplus S_0 \oplus S_1\oplus S_2)\ldots -(x_o\bigoplus_{i=0}^{i=k}S_i)$. Particularly one has $\ell_{X}(2N)=kN=\ell_H(N)$. +(x_o\bigoplus_{i=0}^{i=k}S_i)$. One in particular has $\ell_{X}(2N)=kN=\ell_H(N)$. We claim now that if this PRNG is secure, then the new one is secure too. @@ -1189,8 +1506,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. It follows, by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$ is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}), one has +$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and, +therefore, \begin{equation}\label{PCH-2} -\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1]. +\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1]. \end{equation} Now, using (\ref{PCH-1}) again, one has for every $x$, @@ -1199,7 +1518,7 @@ D^\prime(H(x))=D(\varphi_y(H(x))), \end{equation} where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$, thus -\begin{equation}\label{PCH-3} +\begin{equation}%\label{PCH-3} %%RAPH : j'ai viré ce label qui existe déjà, il est 3 ligne avant D^\prime(H(x))=D(yx), \end{equation} where $y$ is randomly generated. @@ -1209,11 +1528,11 @@ It follows that \mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1]. \end{equation} From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that -there exist a polynomial time probabilistic +there exists a polynomial time probabilistic algorithm $D^\prime$, a positive polynomial $p$, such that for all $k_0$ there exists $N\geq \frac{k_0}{2}$ satisfying $$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$ -proving that $H$ is not secure, a contradiction. +proving that $H$ is not secure, which is a contradiction. \end{proof} @@ -1226,45 +1545,250 @@ It is possible to build a cryptographically secure PRNG based on the previous algorithm (Algorithm~\ref{algo:gpu_kernel2}). Due to Proposition~\ref{cryptopreuve}, it simply consists in replacing the {\it xor-like} PRNG by a cryptographically secure one. -We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form: -$$x_{n+1}=x_n^2~ mod~ M$$ where $M$ is the product of two prime numbers. These -prime numbers need to be congruent to 3 modulus 4. BBS is +We have chosen the Blum Blum Shub generator~\cite{BBS} (usually denoted by BBS) having the form: +$$x_{n+1}=x_n^2~ mod~ M$$ where $M$ is the product of two prime numbers (these +prime numbers need to be congruent to 3 modulus 4). BBS is known to be very slow and only usable for cryptographic applications. -The modulus operation is the most time consuming operation for -current GPU cards. -So in order to obtain quite reasonable performances, it is required -to use only modulus on 32 bits integer numbers. Consequently $x_n^2$ need to be -less than $2^{32}$ and the number $M$ need to be less than $2^{16}$. So in -practice we can choose prime numbers around 256 that are congruent to 3 modulus -4. With 32 bits numbers, only the 4 least significant bits of $x_n$ can be -chosen (the maximum number of indistinguishable bits is lesser than or equals to -$log_2(log_2(x_n))$). So to generate a 32 bits number, we need to use 8 times -the BBS algorithm with different combinations of $M$. +The modulus operation is the most time consuming operation for current +GPU cards. So in order to obtain quite reasonable performances, it is +required to use only modulus on 32-bits integer numbers. Consequently +$x_n^2$ need to be lesser than $2^{32}$, and thus the number $M$ must be +lesser than $2^{16}$. So in practice we can choose prime numbers around +256 that are congruent to 3 modulus 4. With 32-bits numbers, only the +4 least significant bits of $x_n$ can be chosen (the maximum number of +indistinguishable bits is lesser than or equals to +$log_2(log_2(M))$). In other words, to generate a 32-bits number, we need to use +8 times the BBS algorithm with possibly different combinations of $M$. This +approach is not sufficient to be able to pass all the tests of TestU01, +as small values of $M$ for the BBS lead to + small periods. So, in order to add randomness we have proceeded with +the followings modifications. +\begin{itemize} +\item +Firstly, we define 16 arrangement arrays instead of 2 (as described in +Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of +the PRNG kernels. In practice, the selection of combination +arrays to be used is different for all the threads. It is determined +by using the three last bits of two internal variables used by BBS. +%This approach adds more randomness. +In Algorithm~\ref{algo:bbs_gpu}, +character \& is for the bitwise AND. Thus using \&7 with a number +gives the last 3 bits, thus providing a number between 0 and 7. +\item +Secondly, after the generation of the 8 BBS numbers for each thread, we +have a 32-bits number whose period is possibly quite small. So +to add randomness, we generate 4 more BBS numbers to +shift the 32-bits numbers, and add up to 6 new bits. This improvement is +described in Algorithm~\ref{algo:bbs_gpu}. In practice, the last 2 bits +of the first new BBS number are used to make a left shift of at most +3 bits. The last 3 bits of the second new BBS number are added to the +strategy whatever the value of the first left shift. The third and the +fourth new BBS numbers are used similarly to apply a new left shift +and add 3 new bits. +\item +Finally, as we use 8 BBS numbers for each thread, the storage of these +numbers at the end of the kernel is performed using a rotation. So, +internal variable for BBS number 1 is stored in place 2, internal +variable for BBS number 2 is stored in place 3, ..., and finally, internal +variable for BBS number 8 is stored in place 1. +\end{itemize} -Currently this PRNG does not succeed to pass all the tests of TestU01. +\begin{algorithm} +\begin{small} +\KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS +in global memory\; +NumThreads: Number of threads\; +array\_comb: 2D Arrays containing 16 combinations (in first dimension) of size combination\_size (in second dimension)\; +array\_shift[4]=\{0,1,3,7\}\; +} + +\KwOut{NewNb: array containing random numbers in global memory} +\If{threadId is concerned} { + retrieve data from InternalVarBBSArray[threadId] in local variables including shared memory and x\; + we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\; + offset = threadIdx\%combination\_size\; + o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\; + o2 = threadIdx-offset+array\_comb[8+bbs2\&7][offset]\; + \For{i=1 to n} { + t$<<$=4\; + t|=BBS1(bbs1)\&15\; + ...\; + t$<<$=4\; + t|=BBS8(bbs8)\&15\; + \tcp{two new shifts} + shift=BBS3(bbs3)\&3\; + t$<<$=shift\; + t|=BBS1(bbs1)\&array\_shift[shift]\; + shift=BBS7(bbs7)\&3\; + t$<<$=shift\; + t|=BBS2(bbs2)\&array\_shift[shift]\; + t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\; + shared\_mem[threadId]=t\; + x = x\textasciicircum t\; + + store the new PRNG in NewNb[NumThreads*threadId+i]\; + } + store internal variables in InternalVarXorLikeArray[threadId] using a rotation\; +} +\end{small} +\caption{main kernel for the BBS based PRNG GPU} +\label{algo:bbs_gpu} +\end{algorithm} +In Algorithm~\ref{algo:bbs_gpu}, $n$ is for the quantity of random numbers that +a thread has to generate. The operation t<<=4 performs a left shift of 4 bits +on the variable $t$ and stores the result in $t$, and $BBS1(bbs1)\&15$ selects +the last four bits of the result of $BBS1$. Thus an operation of the form +$t<<=4; t|=BBS1(bbs1)\&15\;$ realizes in $t$ a left shift of 4 bits, and then +puts the 4 last bits of $BBS1(bbs1)$ in the four last positions of $t$. Let us +remark that the initialization $t$ is not a necessity as we fill it 4 bits by 4 +bits, until having obtained 32-bits. The two last new shifts are realized in +order to enlarge the small periods of the BBS used here, to introduce a kind of +variability. In these operations, we make twice a left shift of $t$ of \emph{at + most} 3 bits, represented by \texttt{shift} in the algorithm, and we put +\emph{exactly} the \texttt{shift} last bits from a BBS into the \texttt{shift} +last bits of $t$. For this, an array named \texttt{array\_shift}, containing the +correspondence between the shift and the number obtained with \texttt{shift} 1 +to make the \texttt{and} operation is used. For example, with a left shift of 0, +we make an and operation with 0, with a left shift of 3, we make an and +operation with 7 (represented by 111 in binary mode). + +It should be noticed that this generator has once more the form $x^{n+1} = x^n \oplus S^n$, +where $S^n$ is referred in this algorithm as $t$: each iteration of this +PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted +by secure bits produced by the BBS generator, and thus, due to +Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically +secure. -\subsection{A Secure Asymetric Cryptosystem} +\begin{color}{red} +\subsection{Practical Security Evaluation} +Suppose now that the PRNG will work during +$M=100$ time units, and that during this period, +an attacker can realize $10^{12}$ clock cycles. +We thus wonder whether, during the PRNG's +lifetime, the attacker can distinguish this +sequence from truly random one, with a probability +greater than $\varepsilon = 0.2$. +We consider that $N$ has 900 bits. -\section{Conclusion} +The random process is the BBS generator, which +is cryptographically secure. More precisely, it +is $(T,\varepsilon)-$secure: no +$(T,\varepsilon)-$distinguishing attack can be +successfully realized on this PRNG, if~\cite{Fischlin} +$$ +T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M) +$$ +where $M$ is the length of the output ($M=100$ in +our example), and $L(N)$ is equal to +$$ +2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right) +$$ +is the number of clock cycles to factor a $N-$bit +integer. + +A direct numerical application shows that this attacker +cannot achieve its $(10^{12},0.2)$ distinguishing +attack in that context. + +\end{color} +\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem} +\label{Blum-Goldwasser} +We finish this research work by giving some thoughts about the use of +the proposed PRNG in an asymmetric cryptosystem. +This first approach will be further investigated in a future work. -In this paper we have presented a new class of PRNGs based on chaotic -iterations. We have proven that these PRNGs are chaotic in the sense of Devaney. -We also propose a PRNG cryptographically secure and its implementation on GPU. +\subsubsection{Recalls of the Blum-Goldwasser Probabilistic Cryptosystem} + +The Blum-Goldwasser cryptosystem is a cryptographically secure asymmetric key encryption algorithm +proposed in 1984~\cite{Blum:1985:EPP:19478.19501}. The encryption algorithm +implements a XOR-based stream cipher using the BBS PRNG, in order to generate +the keystream. Decryption is done by obtaining the initial seed thanks to +the final state of the BBS generator and the secret key, thus leading to the + reconstruction of the keystream. + +The key generation consists in generating two prime numbers $(p,q)$, +randomly and independently of each other, that are + congruent to 3 mod 4, and to compute the modulus $N=pq$. +The public key is $N$, whereas the secret key is the factorization $(p,q)$. + + +Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Alice: +\begin{enumerate} +\item Bob picks an integer $r$ randomly in the interval $\llbracket 1,N\rrbracket$ and computes $x_0 = r^2~mod~N$. +\item He uses the BBS to generate the keystream of $L$ pseudorandom bits $(b_0, \dots, b_{L-1})$, as follows. For $i=0$ to $L-1$, +\begin{itemize} +\item $i=0$. +\item While $i \leqslant L-1$: +\begin{itemize} +\item Set $b_i$ equal to the least-significant\footnote{As signaled previously, BBS can securely output up to $\mathsf{N} = \lfloor log(log(N)) \rfloor$ of the least-significant bits of $x_i$ during each round.} bit of $x_i$, +\item $i=i+1$, +\item $x_i = (x_{i-1})^2~mod~N.$ +\end{itemize} +\end{itemize} +\item The ciphertext is computed by XORing the plaintext bits $m$ with the keystream: $ c = (c_0, \dots, c_{L-1}) = m \oplus b$. This ciphertext is $[c, y]$, where $y=x_{0}^{2^{L}}~mod~N.$ +\end{enumerate} + + +When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ as follows: +\begin{enumerate} +\item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$. +\item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$. +\item She recomputes the bit-vector $b$ by using BBS and $x_0$. +\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. +\end{enumerate} + + +\subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser} + +We propose to adapt the Blum-Goldwasser protocol as follows. +Let $\mathsf{N} = \lfloor log(log(N)) \rfloor$ be the number of bits that can +be obtained securely with the BBS generator using the public key $N$ of Alice. +Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too, and +her new public key will be $(S^0, N)$. + +To encrypt his message, Bob will compute +%%RAPH : ici, j'ai mis un simple $ +%\begin{equation} +$c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.$ +$ \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)$ +%%\end{equation} +instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. + +The same decryption stage as in Blum-Goldwasser leads to the sequence +$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$. +Thus, with a simple use of $S^0$, Alice can obtain the plaintext. +By doing so, the proposed generator is used in place of BBS, leading to +the inheritance of all the properties presented in this paper. + +\section{Conclusion} -An efficient implementation on GPU based on a xor-like PRNG allows us to -generate a huge number of pseudorandom numbers per second (about -20Gsamples/s). This PRNG succeeds to pass the hardest batteries of TestU01. -In future work we plan to extend this work for parallel PRNG for clusters or -grid computing. We also plan to improve the BBS version in order to succeed all -the tests of TestU01. +In this paper, a formerly proposed PRNG based on chaotic iterations +has been generalized to improve its speed. It has been proven to be +chaotic according to Devaney. +Efficient implementations on GPU using xor-like PRNGs as input generators +have shown that a very large quantity of pseudorandom numbers can be generated per second (about +20Gsamples/s), and that these proposed PRNGs succeed to pass the hardest battery in TestU01, +namely the BigCrush. +Furthermore, we have shown that when the inputted generator is cryptographically +secure, then it is the case too for the PRNG we propose, thus leading to +the possibility to develop fast and secure PRNGs using the GPU architecture. +\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it +behaves chaotically, has finally been proposed. \end{color} + +In future work we plan to extend this research, building a parallel PRNG for clusters or +grid computing. Topological properties of the various proposed generators will be investigated, +and the use of other categories of PRNGs as input will be studied too. The improvement +of Blum-Goldwasser will be deepened. Finally, we +will try to enlarge the quantity of pseudorandom numbers generated per second either +in a simulation context or in a cryptographic one.