X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/5ed99022e098863388088d7f23852378a56cdb5b..8b2ff8fffab74015439e592d520567aec9568d61:/prng_gpu.tex?ds=inline diff --git a/prng_gpu.tex b/prng_gpu.tex index 792d8ce..0a88df5 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -7,6 +7,8 @@ \usepackage{amscd} \usepackage{moreverb} \usepackage{commath} +\usepackage[ruled,vlined]{algorithm2e} +\usepackage{listings} \usepackage[standard]{ntheorem} % Pour mathds : les ensembles IR, IN, etc. @@ -32,399 +34,1426 @@ \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}} -\title{Efficient generation of pseudo random numbers based on chaotic iterations on GPU} +\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU} \begin{document} + +\author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe +Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}} + \maketitle \begin{abstract} -This is the 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 +cards. +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} \section{Introduction} -Interet des itérations chaotiques pour générer des nombre alea\\ -Interet de générer des nombres alea sur GPU -... +Randomness is of importance in many fields 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 +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 +that is usually attained by using parallel architectures. In that case, +a recurrent problem is that a deflate 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 +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 +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 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 +whereas computers deal with finite precision numbers. +This distortion leads to a deflation of both chaotic properties and speed. +Furthermore, authors of such chaotic generators often claim their PRNG +as secure due to their chaos properties, but there is no obvious relation +between chaos and security as it is understood in cryptography. +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 +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}. +On the other hand, chaos can be added to a fast, statistically perfect PRNG and/or a +cryptographically secure one, in case where chaos can be of interest, +\emph{only if these last properties are not lost during +the proposed post-treatment}. Such an assumption is behind this research work. +It leads to the attempts to define a +family of PRNGs that are chaotic while being fast and statistically perfect, +or cryptographically secure. +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}. +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 +family, and that the sequence obtained after this post-treatment can pass the +NIST~\cite{Nist10}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} batteries of tests, even if the inputted generators +cannot. +The proposition of this paper is to improve widely the speed of the formerly +proposed generator, without any lack of chaos or statistical properties. +In particular, a version of this PRNG on graphics processing units (GPU) +is proposed. +Although GPU was initially designed to accelerate +the manipulation of images, they are nowadays commonly used in many scientific +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. +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 rewritten 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. +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 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}, 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} +\label{section:related works} + +Numerous research works on defining GPU based PRNGs have yet been proposed in the +literature, so that completeness 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 +only when the information is given in the related work. +A million numbers per second will be simply written as +1MSample/s whereas a billion numbers per second is 1GSample/s. + +In \cite{Pang:2008:cec} a PRNG based on cellular automata is defined +with no requirement to an high precision integer arithmetic or to any bitwise +operations. Authors can generate about +3.2MSamples/s on a GeForce 7800 GTX GPU, which is quite an old card now. +However, there is neither a mention of statistical tests nor any proof of +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 +CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. +However the evaluations of the proposed PRNGs are only statistical ones. + + +Authors of~\cite{conf/fpga/ThomasHL09} have studied the implementation of some +PRNGs on different computing architectures: CPU, field-programmable gate array +(FPGA), massively parallel processors, and GPU. This study is of interest, because +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 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. + +Lastly, Cuda has developed a library for the generation of pseudorandom numbers called +Curand~\cite{curand11}. Several PRNGs are implemented, among +other things +Xorwow~\cite{Marsaglia2003} and some variants of Sobol. The tests reported show that +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. + +\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. 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 $V_{i}$ +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 +notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$. + + +Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : +\mathcal{X} \rightarrow \mathcal{X}$. -\section{Chaotic iterations} +\begin{definition} +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} -Présentation des itérations chaotiques +\begin{definition} +An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$ +if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$ +\end{definition} +\begin{definition} +$f$ is said to be \emph{regular} on $(\mathcal{X}, \tau)$ if the set of periodic +points for $f$ is dense in $\mathcal{X}$: for any point $x$ in $\mathcal{X}$, +any neighborhood of $x$ contains at least one periodic point (without +necessarily the same period). +\end{definition} +\begin{definition}[Devaney's formulation of chaos~\cite{Devaney}] +The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and +topologically transitive. +\end{definition} -\section{The relativity of disorder} -\label{sec:de la relativité du désordre} +The chaos property is strongly linked to the notion of ``sensitivity'', defined +on a metric space $(\mathcal{X},d)$ by: -\subsection{Impact of the topology's finenesse} +\begin{definition} +\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 $. -Let us firstly introduce the following notations. +The constant $\delta$ is called the \emph{constant of sensitivity} of $f$. +\end{definition} -\begin{notation} -$\mathcal{X}_\tau$ will denote the topological space $\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply $\mathcal{V} (x)$, if there is no ambiguity). -\end{notation} +Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is +chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has the property of +sensitive dependence on initial conditions (this property was formerly an +element of the definition of chaos). To sum up, quoting Devaney +in~\cite{Devaney}, a chaotic dynamical system ``is unpredictable because of the +sensitive dependence on initial conditions. It cannot be broken down or +simplified into two subsystems which do not interact because of topological +transitivity. And in the midst of this random behavior, we nevertheless have an +element of regularity''. Fundamentally different behaviors are consequently +possible and occur in an unpredictable way. -\section{Chaos on the order topology} +\subsection{Chaotic Iterations} +\label{sec:chaotic iterations} -\subsection{The phase space is an interval of the real line} -\subsubsection{Toward a topological semiconjugacy} +Let us consider a \emph{system} with a finite number $\mathsf{N} \in +\mathds{N}^*$ of elements (or \emph{cells}), so that each cell has a +Boolean \emph{state}. Having $\mathsf{N}$ Boolean values for these + cells leads to the definition of a particular \emph{state of the +system}. A sequence which elements belong to $\llbracket 1;\mathsf{N} +\rrbracket $ is called a \emph{strategy}. The set of all strategies is +denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$ -In what follows, our intention is to establish, by using a topological semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as iterations on a real interval. To do so, we must firstly introduce some notations and terminologies. +\begin{definition} +\label{Def:chaotic iterations} +The set $\mathds{B}$ denoting $\{0,1\}$, let +$f:\mathds{B}^{\mathsf{N}}\longrightarrow \mathds{B}^{\mathsf{N}}$ be +a function and $S\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ be a ``strategy''. The so-called +\emph{chaotic iterations} are defined by $x^0\in +\mathds{B}^{\mathsf{N}}$ and +\begin{equation} +\forall n\in \mathds{N}^{\ast }, \forall i\in +\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{ +\begin{array}{ll} + x_i^{n-1} & \text{ if }S^n\neq i \\ + \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i. +\end{array}\right. +\end{equation} +\end{definition} -Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket 1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N} \times \B^\mathsf{N}$. +In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is +\textquotedblleft iterated\textquotedblright . Note that in a more +general formulation, $S^n$ can be a subset of components and +$\left(f(x^{n-1})\right)_{S^{n}}$ can be replaced by +$\left(f(x^{k})\right)_{S^{n}}$, where $k<n$, describing for example, +delays transmission~\cite{Robert1986,guyeux10}. Finally, let us remark that +the term ``chaotic'', in the name of these iterations, has \emph{a +priori} no link with the mathematical theory of chaos, presented above. + + +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} +\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},% +\end{array}% +\end{equation}% +\noindent where + and . are the Boolean addition and product operations. +Consider the phase space: +\begin{equation} +\mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times +\mathds{B}^\mathsf{N}, +\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} +\end{equation} +\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma +(S^{n})_{n\in \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow (S^{n+1})_{n\in +\mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ and $i$ is the \emph{initial function} +$i:(S^{n})_{n\in \mathds{N}} \in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow S^{0}\in \llbracket +1;\mathsf{N}\rrbracket$. Then the chaotic iterations proposed in +Definition \ref{Def:chaotic iterations} can be described by the following iterations: +\begin{equation} +\left\{ +\begin{array}{l} +X^0 \in \mathcal{X} \\ +X^{k+1}=G_{f}(X^k).% +\end{array}% +\right. +\end{equation}% + +With this formulation, a shift function appears as a component of chaotic +iterations. The shift function is a famous example of a chaotic +map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as +chaotic. +To study this claim, a new distance between two points $X = (S,E), Y = +(\check{S},\check{E})\in +\mathcal{X}$ has been introduced in \cite{guyeux10} as follows: +\begin{equation} +d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}), +\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})}, \\ +\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}% +\sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.% +\end{array}% +\right. +\end{equation} -\begin{definition} -The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[ 0, 2^{10} \big[$ is defined by: -$$ -\begin{array}{cccl} -\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}& \longrightarrow & \big[ 0, 2^{10} \big[ \\ - & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto & \varphi \left((S,E)\right) -\end{array} -$$ -\noindent where $\varphi\left((S,E)\right)$ is the real number: +This new distance has been introduced to satisfy the following requirements. \begin{itemize} -\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$. -\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots = \sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ +\item When the number of different cells between two systems is increasing, then +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. \end{itemize} -\end{definition} +The distance presented above follows these recommendations. Indeed, if the floor +value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ +differ in $n$ cells ($d_e$ is indeed the Hamming distance). In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a +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. +Finally, it has been established in \cite{guyeux10} that, +\begin{proposition} +Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. Then $G_{f}$ is continuous in +the metric space $(\mathcal{X},d)$. +\end{proposition} -$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic iterations $\Go$ on this real interval. To do so, two intermediate functions over $\big[ 0, 2^{10} \big[$ must be introduced: +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 +introduced the notion of asynchronous iteration graph recalled bellow. +Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The +{\emph{asynchronous iteration graph}} associated with $f$ is the +directed graph $\Gamma(f)$ defined by: the set of vertices is +$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and +$i\in \llbracket1;\mathsf{N}\rrbracket$, +the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. +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 then proven in \cite{bcgr11:ip} that, -\begin{definition} -\label{def:e et s} -Let $x \in \big[ 0, 2^{10} \big[$ and: -\begin{itemize} -\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$: $\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$. -\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal decomposition of $x$ is the one that does not have an infinite number of 9: -$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$. -\end{itemize} -$e$ and $s$ are thus defined as follows: -$$ -\begin{array}{cccl} -e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\ - & x & \longmapsto & (e_0, \hdots, e_9) -\end{array} -$$ -\noindent and -$$ -\begin{array}{cccl} -s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 \rrbracket^{\mathds{N}} \\ - & x & \longmapsto & (s^k)_{k \in \mathds{N}} -\end{array} -$$ -\end{definition} -We are now able to define the function $g$, whose goal is to translate the chaotic iterations $\Go$ on an interval of $\mathds{R}$. +\begin{theorem} +\label{Th:Caractérisation des IC chaotiques} +Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic (according to Devaney) +if and only if $\Gamma(f)$ is strongly connected. +\end{theorem} -\begin{definition} -$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by: -$$ -\begin{array}{cccl} -g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\ -& \\ - & x & \longmapsto & g(x) +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 lead us to study the possibility to build 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} +\rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$ +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 the two theorems above. + +\section{Application to Pseudorandomness} +\label{sec:pseudorandom} + +\subsection{A First Pseudorandom Number Generator} + +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 +present. + +\begin{algorithm}[h!] +%\begin{scriptsize} +\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)$\; +\For{$i=0,\dots,k$} +{ +$s\leftarrow{\textit{XORshift}(n)}$\; +$x\leftarrow{F_f(s,x)}$\; +} +return $x$\; +%\end{scriptsize} +\caption{PRNG with chaotic functions} +\label{CI Algorithm} +\end{algorithm} + +\begin{algorithm}[h!] +\KwIn{the internal configuration $z$ (a 32-bit word)} +\KwOut{$y$ (a 32-bit word)} +$z\leftarrow{z\oplus{(z\ll13)}}$\; +$z\leftarrow{z\oplus{(z\gg17)}}$\; +$z\leftarrow{z\oplus{(z\ll5)}}$\; +$y\leftarrow{z}$\; +return $y$\; +\medskip +\caption{An arbitrary round of \textit{XORshift} algorithm} +\label{XORshift} +\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 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. + +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. + +\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, +this algorithm can be rewritten as follows: + +\begin{equation} +\left\{ +\begin{array}{l} +x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\ +\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, \end{array} -$$ -\noindent where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow: -\begin{itemize} -\item its integral part has a binary decomposition equal to $e_0', \hdots, e_9'$, with: -$$ -e_i' = \left\{ +\right. +\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 +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 +ordinary use as a good elementary brick in various PRNGs. It corresponds +to the following discrete dynamical system in chaotic iterations: + +\begin{equation} +\forall n\in \mathds{N}^{\ast }, \forall i\in +\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{ \begin{array}{ll} -e(x)_i & \textrm{ if } i \neq s^0\\ -e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\ -\end{array} + 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. +\end{array}\right. +\label{eq:generalIC} +\end{equation} +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, +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 +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 +use of more general chaotic iterations to generate pseudorandom numbers +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: + +\begin{equation} +\forall n\in \mathds{N}^{\ast }, \forall i\in +\llbracket1;\mathsf{N}\rrbracket ,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. +\end{array}\right. +\label{general CIs} +\end{equation} + +In other words, at the $n^{th}$ iteration, only the cells whose id is +contained into the set $S^{n}$ are iterated. + +Let us now rewrite these general chaotic iterations as usual discrete dynamical +system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation +is required in order to study the topological behavior of the system. + +Let us introduce the following function: +\begin{equation} +\begin{array}{cccc} + \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\ + & (i,X) & \longmapsto & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X, \end{array}\right. +\end{array} +\end{equation} +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},% +\end{array}% +\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: +\begin{equation} +\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times +\mathds{B}^\mathsf{N}, +\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} +\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 +\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function} +$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$. +Then the general chaotic iterations defined in Equation \ref{general CIs} can +be described by the following discrete dynamical system: +\begin{equation} +\left\{ +\begin{array}{l} +X^0 \in \mathcal{X} \\ +X^{k+1}=G_{f}(X^k).% +\end{array}% \right. -$$ -\item whose decimal part is $s(x)^1, s(x)^2, \hdots$ -\end{itemize} -\end{definition} +\end{equation}% + +Another time, a shift function appears as a component of these general chaotic +iterations. + +To study the Devaney's chaos property, a distance between two points +$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined. +Let us introduce: +\begin{equation} +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} +where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as +$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$. -\bigskip +\begin{proposition} +The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$. +\end{proposition} + +\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. + \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 +$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$. + \item $d_s$ is symmetric +($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property +of the symmetric difference. +\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, +and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, +we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle +inequality is obtained. + \end{itemize} +\end{proof} -In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k + \sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then: $$g(x) = \displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + \sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}.$$ -\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$} +Before being able to study the topological behavior of the general +chaotic iterations, we must firstly establish that: -Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most usual one being the Euclidian distance recalled bellow: +\begin{proposition} + For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on +$\left( \mathcal{X},d\right)$. +\end{proposition} -\begin{notation} -\index{distance!euclidienne} -$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is, $\Delta(x,y) = |y-x|^2$. -\end{notation} +\begin{proof} +We use the sequential continuity. +Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $% +\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left( +G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left( +G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy, +thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of +sequences).\newline +As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges +to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $% +d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline +In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no +cell will change its state: +$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$ + +In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in % +\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $% +n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same +first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$ + +Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are +identical and strategies $S^n$ and $S$ start with the same first term.\newline +Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal, +so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline +\noindent We now prove that the distance between $\left( +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 +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 +\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant +\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so +\begin{equation*} +\exists n_{2}\in \mathds{N},\forall n\geqslant +n_{2},d_{s}(S^n,S)<10^{-(k+2)}, +\end{equation*}% +thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal. +\end{itemize} +\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 +In conclusion, +$$ +\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}% +,\forall n\geqslant N_{0}, + d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) +\leqslant \varepsilon . +$$ +$G_{f}$ is consequently continuous. +\end{proof} -This Euclidian distance does not reproduce exactly the notion of proximity induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$. This is the reason why we have to introduce the following metric: +It is now possible to study the topological behavior of the general chaotic +iterations. We will prove that, +\begin{theorem} +\label{t:chaos des general} + The general chaotic iterations defined on Equation~\ref{general CIs} satisfy +the Devaney's property of chaos. +\end{theorem} -\begin{definition} -Let $x,y \in \big[ 0, 2^{10} \big[$. -$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$ defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$, where: -\begin{center} -$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k, \check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$. -\end{center} -\end{definition} +Let us firstly prove the following lemma. -\begin{proposition} -$D$ is a distance on $\big[ 0, 2^{10} \big[$. -\end{proposition} +\begin{lemma}[Strong transitivity] +\label{strongTrans} + For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can +find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$. +\end{lemma} \begin{proof} -The three axioms defining a distance must be checked. + Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. +Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, +are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define +$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$. +We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates +that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of +the form $(S',E')$ where $E'=E$ and $S'$ starts with +$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties: \begin{itemize} -\item $D \geqslant 0$, because everything is positive in its definition. If $D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal (they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then $\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$. -\item $D(x,y)=D(y,x)$. -\item Finally, the triangular inequality is obtained due to the fact that both $\delta$ and $\Delta(x,y)=|x-y|$ satisfy it. + \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$, + \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$. \end{itemize} +Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, +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}. + +\begin{proof}[Theorem~\ref{t:chaos des general}] +Firstly, strong transitivity implies transitivity. + +Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To +prove that $G_f$ is regular, it is sufficient to prove that +there exists a strategy $\tilde S$ such that the distance between +$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that +$(\tilde S,E)$ is a periodic point. + +Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the +configuration that we obtain from $(S,E)$ after $t_1$ iterations of +$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ +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 +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 $t<t_1$, by the choice of $t_1$, we +have $d((S,E),(\tilde S,E))<\epsilon$. +\end{proof} -The convergence of sequences according to $D$ is not the same than the usual convergence related to the Euclidian metric. For instance, if $x^n \to x$ according to $D$, then necessarily the integral part of each $x^n$ is equal to the integral part of $x$ (at least after a given threshold), and the decimal part of $x^n$ corresponds to the one of $x$ ``as far as required''. -To illustrate this fact, a comparison between $D$ and the Euclidian distance is given Figure \ref{fig:comparaison de distances}. These illustrations show that $D$ is richer and more refined than the Euclidian distance, and thus is more precise. -\begin{figure}[t] -\begin{center} - \subfigure[Function $x \to dist(x;1,234) $ on the interval $(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad - \subfigure[Function $x \to dist(x;3) $ on the interval $(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}} -\end{center} -\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).} -\label{fig:comparaison de distances} -\end{figure} +\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 $x \oplus S^i$. -\subsubsection{The semiconjugacy} +\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} +\end{table} + + + + +\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iteration\ +s},label=algo:seqCIPRNG} +\begin{lstlisting} +unsigned int CIPRNG() { + static unsigned int x = 123123123; + unsigned long t1 = xorshift(); + unsigned long t2 = xor128(); + unsigned long t3 = xorwow(); + x = x^(unsigned int)t1; + x = x^(unsigned int)(t2>>32); + x = x^(unsigned int)(t3>>32); + x = x^(unsigned int)t2; + x = x^(unsigned int)(t1>>32); + x = x^(unsigned int)t3; + return x; +} +\end{lstlisting} -It is now possible to define a topological semiconjugacy between $\mathcal{X}$ and an interval of $\mathds{R}$: -\begin{theorem} -Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on $\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow: -\begin{equation*} -\begin{CD} -\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>> \left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\ - @V{\varphi}VV @VV{\varphi}V\\ -\left( ~\big[ 0, 2^{10} \big[, D~\right) @>>g> \left(~\big[ 0, 2^{10} \big[, D~\right) -\end{CD} -\end{equation*} -\end{theorem} -\begin{proof} -$\varphi$ has been constructed in order to be continuous and onto. -\end{proof} -In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} \big[$. +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}. +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 +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 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} -\subsection{Study of the chaotic iterations described as a real function} + +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. +Of course, the three xor-like +PRNGs used in these computations must have different parameters. +In a given thread, these lasts 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 +parameters embedded into each thread. + +The implementation of the three +xor-like PRNGs is straightforward when their parameters have been +allocated in the GPU memory. Each xor-like works with an internal +number $x$ that saves the last generated pseudorandom number. Additionally, the +implementation of the xor128, the xorshift, and the xorwow respectively require +4, 5, and 6 unsigned long as internal variables. + +\begin{algorithm} + +\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like +PRNGs in global memory\; +NumThreads: number of threads\;} +\KwOut{NewNb: array containing random numbers in global memory} +\If{threadIdx is concerned by the computation} { + retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\; + \For{i=1 to n} { + compute a new PRNG as in Listing\ref{algo:seqCIPRNG}\; + store the new PRNG in NewNb[NumThreads*threadIdx+i]\; + } + store internal variables in InternalVarXorLikeArray[threadIdx]\; +} +\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 +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)}, +then the memory required to store all of the internals variables of both the xor-like +PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers} +and the pseudorandom numbers generated by our PRNG, is equal to $100,000\times ((4+5+6)\times +2+(1+100))=1,310,000$ 32-bits numbers, that is, approximately $52$Mb. + +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). -\begin{figure}[t] -\begin{center} - \subfigure[ICs on the interval $(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad - \subfigure[ICs on the interval $(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\ - \subfigure[ICs on the interval $(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad - \subfigure[ICs on the interval $(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}} -\end{center} -\caption{Representation of the chaotic iterations.} -\label{fig:ICs} -\end{figure} +\begin{remark} +The proposed algorithm has the advantage to manipulate 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. +\end{remark} +\subsection{Improved Version for GPU} + +As GPU cards using CUDA have shared memory between threads of the same block, it +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 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 +\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. + +\begin{algorithm} + +\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs +in global memory\; +NumThreads: Number of threads\; +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+array\_comb1[offset]\; + o2 = threadIdx-offset+array\_comb2[offset]\; + \For{i=1 to n} { + t=xor-like()\; + 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]\; +} +\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} + +A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having +the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative +system of Eq.~\ref{eq:generalIC}. That is, an iteration of the general chaotic +iterations is realized between the last stored value $x$ of the thread and a strategy $t$ +(obtained by a bitwise exclusive or between a value provided by a xor-like() call +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 $\mathds{B}^ \mathsf{N}$. +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 $\llbracket 1, \mathsf{N} \rrbracket$. + +Such a result is obvious, as for the xor-like(), all the +integers belonging into its interval of definition can occur at each iteration, and thus the +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 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 +Devaney's formulation of a chaotic behavior. +\section{Experiments} +\label{sec:experiments} + +Different experiments have been performed in order to measure the generation +speed. We have used a first computer equipped with a Tesla C1060 NVidia GPU card +and an +Intel Xeon E5530 cadenced at 2.40 GHz, and +a second computer equipped with a smaller CPU and a GeForce GTX 280. +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 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 +into the GPU memory has been removed. This step is time consuming and slows down the numbers +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 +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 +practice, the Tesla C1060 has more memory than the GTX 280, and this memory +should be of better quality. +As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of about +138MSample/s when using one core of the Xeon E5530. -\begin{figure}[t] +\begin{figure}[htbp] \begin{center} - \subfigure[ICs on the interval $(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad - \subfigure[ICs on the interval $(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}} + \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf} \end{center} -\caption{ICs on small intervals.} -\label{fig:ICs2} +\caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG} +\label{fig:time_xorlike_gpu} \end{figure} -\begin{figure}[t] -\begin{center} - \subfigure[ICs on the interval $(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad - \subfigure[ICs on the interval $(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad -\end{center} -\caption{General aspect of the chaotic iterations.} -\label{fig:ICs3} -\end{figure} -We have written a Python program to represent the chaotic iterations with the vectorial negation on the real line $\mathds{R}$. Various representations of these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}. It can be remarked that the function $g$ is a piecewise linear function: it is linear on each interval having the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its slope is equal to 10. Let us justify these claims: -\begin{proposition} -\label{Prop:derivabilite des ICs} -Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on $\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$. -Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, $g$ is a linear function, having a slope equal to 10: $\forall x \notin I, g'(x)=10$. -\end{proposition} +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. +\begin{figure}[htbp] +\begin{center} + \includegraphics[scale=.7]{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} +\end{figure} -\begin{proof} -Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral prat $e$ and the same decimal part $s^0$: on the set $I_n$, functions $e(x)$ and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all the images $g(x)$ of these points $x$: -\begin{itemize} -\item Have the same integral part, which is $e$, except probably the bit number $s^0$. In other words, this integer has approximately the same binary decomposition than $e$, the sole exception being the digit $s^0$ (this number is then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$, \emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$). -\item A shift to the left has been applied to the decimal part $y$, losing by doing so the common first digit $s^0$. In other words, $y$ has been mapped into $10\times y - s^0$. -\end{itemize} -To sum up, the action of $g$ on the points of $I$ is as follows: first, make a multiplication by 10, and second, add the same constant to each term, which is $\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$. -\end{proof} +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 +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. -\begin{remark} -Finally, chaotic iterations are elements of the large family of functions that are both chaotic and piecewise linear (like the tent map). -\end{remark} -\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$} -The two propositions bellow allow to compare our two distances on $\big[ 0, 2^\mathsf{N} \big[$: -\begin{proposition} -Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0, 2^\mathsf{N} \big[, D~\right)$ is not continuous. -\end{proposition} -\begin{proof} -The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is such that: -\begin{itemize} -\item $\Delta (x^n,2) \to 0.$ -\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0. -\end{itemize} +\section{Security Analysis} +\label{sec:security analysis} -The sequential characterization of the continuity concludes the demonstration. -\end{proof} +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 $m$ of length $m$, $G(m)$ (the output of $G$ on the input $m$) has size +$\ell_G(m)$ with $\ell_G(m)>m$. +The notion of {\it secure} PRNGs can now be defined as follows. -A contrario: +\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 $m$'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 $U_m$, $U_{\ell_G(m)}$ as well as over the +internal coin tosses of $D$. +\end{definition} + +Intuitively, it means that there is no polynomial time algorithm that can +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 $\ell^\prime(m)>m)$~\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, +without loss of generality, that for any string $S_0$ of size $N$, the size +of $H(S_0)$ is $kN$, with $k>2$. It means that $\ell_H(N)=kN$. +Let $S_1,\ldots,S_k$ be the +strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concatenation of +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 $\ell_{X}(2N)=kN=\ell_H(N)$. +We claim now that if this PRNG is secure, +then the new one is secure too. \begin{proposition} -Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0, 2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction. +\label{cryptopreuve} +If $H$ is a secure cryptographic PRNG, then $X$ is a secure cryptographic +PRNG too. \end{proposition} \begin{proof} -If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given threshold, because $D_e$ only returns integers. So, after this threshold, the integral parts of all the $x^n$ are equal to the integral part of $x$. - -Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k \in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This means that for all $k$, an index $N_k$ can be found such that, $\forall n \geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the result. +The proposition is proved by contraposition. Assume that $X$ is not +secure. By Definition, there exists a polynomial time probabilistic +algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists +$N\geq \frac{k_0}{2}$ satisfying +$$| \mathrm{Pr}[D(X(U_{2N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)}.$$ +We describe a new probabilistic algorithm $D^\prime$ on an input $w$ of size +$kN$: +\begin{enumerate} +\item Decompose $w$ into $w=w_1\ldots w_{k}$, where each $w_i$ has size $N$. +\item Pick a string $y$ of size $N$ uniformly at random. +\item Compute $z=(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y + \bigoplus_{i=1}^{i=k} w_i).$ +\item Return $D(z)$. +\end{enumerate} + + +Consider for each $y\in \mathbb{B}^{kN}$ the function $\varphi_{y}$ +from $\mathbb{B}^{kN}$ into $\mathbb{B}^{kN}$ mapping $w=w_1\ldots w_k$ +(each $w_i$ has length $N$) to +$(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y + \bigoplus_{i=1}^{i=k_1} w_i).$ By construction, one has for every $w$, +\begin{equation}\label{PCH-1} +D^\prime(w)=D(\varphi_y(w)), +\end{equation} +where $y$ is randomly generated. +Moreover, for each $y$, $\varphi_{y}$ is injective: if +$(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y\bigoplus_{i=1}^{i=k_1} +w_i)=(y\oplus w_1^\prime)(y\oplus w_1^\prime\oplus w_2^\prime)\ldots +(y\bigoplus_{i=1}^{i=k} w_i^\prime)$, then for every $1\leq j\leq k$, +$y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. 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 +\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]. +\end{equation} + +Now, using (\ref{PCH-1}) again, one has for every $x$, +\begin{equation}\label{PCH-3} +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} +D^\prime(H(x))=D(yx), +\end{equation} +where $y$ is randomly generated. +It follows that + +\begin{equation}\label{PCH-4} +\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 +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. \end{proof} -The conclusion of these propositions is that the proposed metric is more precise than the Euclidian distance, that is: - -\begin{corollary} -$D$ is finer than the Euclidian distance $\Delta$. -\end{corollary} - -This corollary can be reformulated as follows: +\section{Cryptographical Applications} + +\subsection{A Cryptographically Secure PRNG for GPU} +\label{sec:CSGPU} + +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 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 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 TestU01, +as small values of $M$ for the BBS lead to + small periods. So, in order to add randomness we proceed with +the followings modifications. \begin{itemize} -\item The topology produced by $\Delta$ is a subset of the topology produced by $D$. -\item $D$ has more open sets than $\Delta$. -\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than to converge with the one inherited by $\Delta$, which is denoted here by $\tau_\Delta$. +\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 combinations +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, providing so 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 add 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} +\begin{algorithm} -\subsection{Chaos of the chaotic iterations on $\mathds{R}$} -\label{chpt:Chaos des itérations chaotiques sur R} - - +\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\}\; +} -\subsubsection{Chaos according to Devaney} +\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\; +} -We have recalled previously that the chaotic iterations $\left(\Go, \mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We can deduce that they are chaotic on $\mathds{R}$ too, when considering the order topology, because: +\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 +correspondance 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} +\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. + +\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 $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10} \big[_D\right)$ are semiconjugate by $\varphi$, -\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic according to Devaney, because the semiconjugacy preserve this character. -\item But the topology generated by $D$ is finer than the topology generated by the Euclidian distance $\Delta$ -- which is the order topology. -\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order topology on $\mathds{R}$. +\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} - -This result can be formulated as follows. - -\begin{theorem} -\label{th:IC et topologie de l'ordre} -The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the order topology. -\end{theorem} - -Indeed this result is weaker than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary. - +\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 computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. +\end{enumerate} -\section{Efficient prng based on chaotic iterations} -On parle du séquentiel avec des nombres 64 bits\\ +\subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser} -Faire le lien avec le paragraphe précédent (je considère que la stratégie s'appelle $S^i$\\ +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 $(S^0, N)$. -In order to implement efficiently a PRNG based on chaotic iterations it is -possible to improve previous works [ref]. One solution consists in considering -that the strategy used $S^i$ contains all the bits for which the negation is -achieved out. Then instead of applying the negation on these bits we can simply -apply the xor operator between the current number and the strategy $S^i$. In -order to obtain the strategy we also use a classical PRNG. +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} +instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. -\begin{figure}[htbp] -\begin{center} -\fbox{ -\begin{minipage}{14cm} -unsigned int CIprng() \{\\ - static unsigned int x = 123123123;\\ - unsigned long t1 = xorshift();\\ - unsigned long t2 = xor128();\\ - unsigned long t3 = xorwow();\\ - x = x\textasciicircum (unsigned int)t1;\\ - x = x\textasciicircum (unsigned int)(t2$>>$32);\\ - x = x\textasciicircum (unsigned int)(t3$>>$32);\\ - x = x\textasciicircum (unsigned int)t2;\\ - x = x\textasciicircum (unsigned int)(t1$>>$32);\\ - x = x\textasciicircum (unsigned int)t3;\\ - return x;\\ -\} -\end{minipage} -} -\end{center} -\caption{sequential Chaotic Iteration PRNG} -\label{algo:seqCIprng} -\end{figure} +The same decryption stage as in Blum-Goldwasser leads to the sequence +$\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. +By doing so, the proposed generator is used in place of BBS, leading to +the inheritance of all the properties presented in this paper. -In Figure~\ref{algo:seqCIprng} a sequential version of our chaotic iterations -based PRNG is presented. This version uses three classical 64 bits PRNG: the -\texttt{xorshift}, the \texttt{xor128} and the \texttt{xorwow}. These three -PRNGs are presented in~\cite{Marsaglia2003}. As each PRNG used works with -64-bits and as our PRNG works with 32 bits, the use of \texttt{(unsigned int)} -selects the 32 least significant bits whereas \texttt{(unsigned int)(t3$>>$32)} -selects the 32 most significants bits of the variable \texttt{t}. This version -sucesses the BigCrush of the TestU01 battery [P. Lâecuyer and - R. Simard. Testu01]. +\section{Conclusion} -\section{Efficient prng based on chaotic iterations on GPU} -On parle du passage du sequentiel au GPU +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 +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. +Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the +proposed method, has been finally proposed. -\section{Experiments} +In future work we plan to extend these researches, 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. -On passe le BigCrush\\ -On donne des temps de générations sur GPU/CPU\\ -On donne des temps de générations de nombre sur GPU puis on rappatrie sur CPU / CPU ? bof bof, on verra -\section{Conclusion} -\bibliographystyle{plain} +\bibliographystyle{plain} \bibliography{mabase} \end{document}