X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/4635b3f66f097ccf0ad342948e2a29906bdbb32a..26a94eb935af7804c64342d5c4718e05c9e10036:/prng_gpu.tex?ds=inline diff --git a/prng_gpu.tex b/prng_gpu.tex index 6782175..983aa92 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 +8,14 @@ \usepackage{amscd} \usepackage{moreverb} \usepackage{commath} -\usepackage{algorithm2e} +\usepackage[ruled,vlined]{algorithm2e} \usepackage{listings} \usepackage[standard]{ntheorem} +\usepackage{algorithmic} +\usepackage{slashbox} +\usepackage{ctable} +\usepackage{tabularx} +\usepackage{multirow} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -38,27 +44,35 @@ \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 +80,27 @@ 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 +sequence. \begin{color}{red} Or, in an equivalent formulation, he or she should not be +able (in practice) to predict the next bit of the generator, having the knowledge of all the +binary digits that have been already released. ``Being able in practice'' refers here +to the possibility to achieve this attack in polynomial time, and to the exponential growth +of the difficulty of this challenge when the size of the parameters of the PRNG increases. +\end{color} + +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 +114,7 @@ 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}. @@ -109,11 +129,20 @@ 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}. +\begin{color}{red} +More precisely, each time we performed a test on a PRNG, we ran it +twice in order to observe if all $p-$values are inside [0.01, 0.99]. In +fact, we observed that few $p-$values (less than ten) are sometimes +outside this interval but inside [0.001, 0.999], so that is why a +second run allows us to confirm that the values outside are not for +the same test. With this approach all our PRNGs pass the {\it + BigCrush} successfully and all $p-$values are at least once inside +[0.01, 0.99]. +\end{color} Chaos, for its part, refers to the well-established definition of a chaotic dynamical system proposed by Devaney~\cite{Devaney}. - In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave as a chaotic dynamical system. Such a post-treatment leads to a new category of PRNGs. We have shown that proofs of Devaney's chaos can be established for this @@ -130,39 +159,48 @@ 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}. -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. +The proposed PRNG and its proof of chaos are given in Section~\ref{sec:pseudorandom}. +\begin{color}{red} +Section~\ref{The generation of pseudorandom sequence} illustrates the statistical +improvement related to the chaotic iteration based post-treatment, for +our previously released PRNGs and a new efficient +implementation on CPU. +\end{color} + Section~\ref{sec:efficient PRNG + 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}, \begin{color}{red} to a practical +security evaluation in Section~\ref{sec:Practicak evaluation}, \end{color} 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. -\section{Related works on GPU based PRNGs} +\section{Related work 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 +218,7 @@ 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 +230,11 @@ 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 @@ -206,15 +244,18 @@ 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. -\subsection{Devaney's Chaotic Dynamical Systems} +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} +\label{subsec:Devaney} In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$ is for the $k^{th}$ composition of a function $f$. Finally, the following @@ -225,7 +266,7 @@ 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 +285,7 @@ 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 +293,12 @@ 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 +357,15 @@ 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 +421,9 @@ 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 +432,7 @@ 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 +442,7 @@ 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 +455,7 @@ 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 +464,33 @@ 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 +499,60 @@ $\mathds{B}^\mathsf{N}$ 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 improve 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 +560,100 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\; $z\leftarrow{z\oplus{(z\ll5)}}$\; $y\leftarrow{z}$\; 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 pseudorandom 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 pseudorandom 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 must be chosen such that the outputs of the resulted PRNG are uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this +goal (other candidates and more information can be found in ~\cite{bg10:ip}). +\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$k\leftarrow{g(a)}$\; +\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} @@ -526,7 +666,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \label{equation Oplus} \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 @@ -550,14 +690,16 @@ 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 Oplus}, 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 +709,11 @@ 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 +738,13 @@ 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 +754,7 @@ 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 +771,7 @@ 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 +781,21 @@ 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 +806,7 @@ 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 +823,7 @@ 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 +859,7 @@ 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 +874,16 @@ 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 +923,7 @@ 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 +941,10 @@ 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 $t0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillates as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according +to Li-Yorke~\cite{Li75,Ruette2001}. This property is related to the following test~\cite{Nist10}. + \begin{itemize} + \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow. + \end{itemize} + \item \textbf{Topological entropy}. Both in topological and statistical fields. + \begin{itemize} +\item \textbf{Approximate Entropy Test}. Compare the frequency of overlapping blocks of two consecutive/adjacent lengths (m and m+1) against the expected result for a random sequence (m is the length of each block). + \end{itemize} + + \item \textbf{Non-linearity, complexity}. + \begin{itemize} +\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence. +\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random (M is the length in bits of a block). + \end{itemize} +\end{itemize} + + + + + +\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 generators (LCGs) will be used. +They are 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, 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 experiments. + +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 (INV) generator~\cite{LEcuyerS07} has been studied, 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} -\section{Efficient PRNG based on Chaotic Iterations} -\label{sec:efficient PRNG} -Based on the proof presented in the previous section, it is now possible to -improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. -The first idea is to consider -that the provided strategy is a pseudorandom Boolean vector obtained by a -given PRNG. -An iteration of the system is simply the bitwise exclusive or between -the last computed state and the current strategy. -Topological properties of disorder exhibited by chaotic -iterations can be inherited by the inputted generator, hoping by doing so to -obtain some statistical improvements while preserving speed. - - -Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations -are -done. -Suppose that $x$ and the strategy $S^i$ are given as -binary vectors. -Table~\ref{TableExemple} shows the result of $x \oplus S^i$. \begin{table} -$$ -\begin{array}{|cc|cccccccccccccccc|} -\hline -x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\ -\hline -S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\ -\hline -x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\ -\hline - -\hline - \end{array} -$$ -\caption{Example of an arbitrary round of the proposed generator} -\label{TableExemple} +\renewcommand{\arraystretch}{1.3} +\caption{TestU01 Statistical Test} +\label{TestU011} +\centering + \begin{tabular}{lccccc} + \toprule +Test name &Tests& Logistic & XORshift & ISAAC\\ +Rabbit & 38 &21 &14 &0 \\ +Alphabit & 17 &16 &9 &0 \\ +Pseudo DieHARD &126 &0 &2 &0 \\ +FIPS\_140\_2 &16 &0 &0 &0 \\ +SmallCrush &15 &4 &5 &0 \\ +Crush &144 &95 &57 &0 \\ +Big Crush &160 &125 &55 &0 \\ \hline +Failures & &261 &146 &0 \\ +\bottomrule + \end{tabular} \end{table} -\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG} +\begin{table} +\renewcommand{\arraystretch}{1.3} +\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)} +\label{TestU01 for Old CI} +\centering + \begin{tabular}{lcccc} + \toprule +\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\ +&Logistic& XORshift& ISAAC&ISAAC \\ +&+& +& + & + \\ +&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5} +Rabbit &7 &2 &0 &0 \\ +Alphabit & 3 &0 &0 &0 \\ +DieHARD &0 &0 &0 &0 \\ +FIPS\_140\_2 &0 &0 &0 &0 \\ +SmallCrush &2 &0 &0 &0 \\ +Crush &47 &4 &0 &0 \\ +Big Crush &79 &3 &0 &0 \\ \hline +Failures &138 &9 &0 &0 \\ +\bottomrule + \end{tabular} +\end{table} + + + + + +\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 score is less spectacular: a large speed improvement makes that statistics + are not as good as for the two other versions of these CIPRNGs. +However 8 tests have been improved (with no deflation for the other results). + + +\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 Table~\ref{threshold}, illustrating +the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in. +Thus 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*} + +Finally, the TestU01 battery has been launched on three well-known generators +(a logistic map, a simple XORshift, and the cryptographically secure ISAAC, +see Table~\ref{TestU011}). These results can be compared with +Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the +Old CI PRNG that has received these generators. +The obvious improvement speaks for itself, and together with the other +results recalled in this section, it reinforces the opinion that a strong +correlation between topological properties and statistical behavior exists. + + +Next subsection will now give a concrete original implementation of the Xor CI PRNG, the +fastest generator in the chaotic iteration based family. In the remainder, +this generator will be simply referred as CIPRNG, or ``the proposed PRNG'', if this statement does not +raise ambiguity. +\end{color} + +\subsection{Efficient Implementation of a PRNG based on Chaotic Iterations} +\label{sec:efficient PRNG} +% +%Based on the proof presented in the previous section, it is now possible to +%improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. +%The first idea is to consider +%that the provided strategy is a pseudorandom Boolean vector obtained by a +%given PRNG. +%An iteration of the system is simply the bitwise exclusive or between +%the last computed state and the current strategy. +%Topological properties of disorder exhibited by chaotic +%iterations can be inherited by the inputted generator, we hope by doing so to +%obtain some statistical improvements while preserving speed. +% +%%RAPH : j'ai viré tout ca +%% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations +%% are +%% done. +%% Suppose that $x$ and the strategy $S^i$ are given as +%% binary vectors. +%% Table~\ref{TableExemple} shows the result of $x \oplus S^i$. + +%% \begin{table} +%% \begin{scriptsize} +%% $$ +%% \begin{array}{|cc|cccccccccccccccc|} +%% \hline +%% x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\ +%% \hline +%% S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\ +%% \hline +%% x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\ +%% \hline + +%% \hline +%% \end{array} +%% $$ +%% \end{scriptsize} +%% \caption{Example of an arbitrary round of the proposed generator} +%% \label{TableExemple} +%% \end{table} + + + + +\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}} +\begin{small} \begin{lstlisting} + unsigned int CIPRNG() { static unsigned int x = 123123123; unsigned long t1 = xorshift(); @@ -862,23 +1324,22 @@ unsigned int CIPRNG() { return x; } \end{lstlisting} +\end{small} +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)(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} @@ -890,12 +1351,12 @@ 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 +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. With CUDA parts of the code which are executed by the GPU are +them. Furthermore, in CUDA, parts of the code that are executed by the GPU, are called {\it kernels}. @@ -903,10 +1364,10 @@ called {\it kernels}. 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 @@ -919,8 +1380,9 @@ 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\;} @@ -933,14 +1395,16 @@ 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)}, @@ -951,14 +1415,14 @@ 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} @@ -972,42 +1436,44 @@ 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 combination arrays are used. -The variable \texttt{offset} is computed using the value of +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 64-bits xor-like -PRNG has been chosen, and so its two 32-bits parts are used. +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 combinations of size combination\_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\%combination\_size\; - o1 = threadIdx-offset+tab1[offset]\; - o2 = threadIdx-offset+tab2[offset]\; + o1 = threadIdx-offset+array\_comb1[offset]\; + o2 = threadIdx-offset+array\_comb2[offset]\; \For{i=1 to n} { t=xor-like()\; - t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\; + t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\; shared\_mem[threadId]=t\; - x = x $\hat{ }$ 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} @@ -1021,7 +1487,7 @@ 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 $\llbracket 1, \mathsf{N} \rrbracket$. @@ -1032,8 +1498,7 @@ 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 @@ -1051,7 +1516,7 @@ 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 @@ -1059,7 +1524,7 @@ 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 @@ -1070,7 +1535,7 @@ 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} @@ -1080,17 +1545,16 @@ 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 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 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} @@ -1098,7 +1562,7 @@ 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. @@ -1118,17 +1582,17 @@ 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 $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size -$\ell_G(k)$ with $\ell_G(k)>k$. +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, -$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$ +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} @@ -1137,7 +1601,7 @@ 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, @@ -1148,7 +1612,7 @@ 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. @@ -1192,8 +1656,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. 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 $x$, @@ -1202,7 +1668,7 @@ 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. @@ -1212,11 +1678,11 @@ 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} @@ -1229,105 +1695,162 @@ 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 +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 -$log_2(log_2(x_n))$). So to generate a 32 bits number, we need to use -8 times the BBS algorithm with different combinations of $M$. This -approach is not sufficient to pass all the tests of TestU01 because -the fact of having chosen small values of $M$ for the BBS leads to -have a small period. So, in order to add randomness we proceed with +$log_2(log_2(M))$). 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 -First we define 16 arrangement arrays instead of 2 (as described in -algorithm \ref{algo:gpu_kernel2}) but only 2 are used at each call of -the PRNG kernels. In practice, the selection of which combinations -arrays will be used is different for all the threads and is determined +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 \& performs the AND bitwise. So using \&7 with a number -gives the last 3 bits, so it provides a number between 0 and 7. +%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 -Second, after the generation of the 8 BBS numbers for each thread we -have a 32 bits number for which the period is possibly quite small. So -to add randomness, we generate 4 more BBS numbers which allows us to -shift the 32 bits numbers and add upto 6 new bits. This part is -described in algorithm~\ref{algo:bbs_gpu}. In practice, if we call -{\it strategy}, the number representing the strategy, the last 2 bits -of the first new BBS number are used to make a left shift of at least -3 bits. The last 3 bits of the second new BBS number are add to the +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 store of these +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 store ind place 3, ... and 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} - \begin{algorithm} - +\begin{small} \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS in global memory\; NumThreads: Number of threads\; -tab: 2D Arrays containing 16 combinations (in first dimension) of size combination\_size (in second dimension)\;} +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+tab[bbs1\&7][offset]\; - o2 = threadIdx-offset+tab[8+bbs2\&7][offset]\; + 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$<<$=4\; t|=BBS1(bbs1)\&15\; ...\; - t<<=4\; + t$<<$=4\; t|=BBS8(bbs8)\&15\; - //two new shifts\; - t<<=BBS3(bbs3)\&3\; - t|=BBS1(bbs1)\&7\; - t<<=BBS7(bbs7)\&3\; - t|=BBS2(bbs2)\&7\; - t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\; + \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 $\hat{ }$ 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}, t<<=4 performs a left shift of 4 bits -on the variable t and stores the result in t. BBS1(bbs1)\&15 selects -the last four bits of the result of BBS1. It should be noticed that -for the two new shifts, we use arbitrarily 4 BBSs that have previously -been used. +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{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem} +\begin{color}{red} +\subsection{Practical Security Evaluation} +\label{sec:Practicak 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. +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. @@ -1355,7 +1878,7 @@ Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Ali \item $i=0$. \item While $i \leqslant L-1$: \begin{itemize} -\item Set $b_i$ equal to the least-significant\footnote{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 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} @@ -1369,7 +1892,7 @@ When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ \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 computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. +\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. \end{enumerate} @@ -1382,30 +1905,41 @@ Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too her new public key will be $(S^0, N)$. To encrypt his message, Bob will compute -\begin{equation} -c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right) -\end{equation} +%%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 obtained the plaintext. +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} -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. - -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. +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.