X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/77973c61177ccd9f68459a2095c29d48d11af26f..559dbc1685c4f30e9ff30ebb4609965f421c697c:/prng_gpu.tex?ds=sidebyside diff --git a/prng_gpu.tex b/prng_gpu.tex index e177e24..5b07118 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -4,83 +4,1510 @@ \usepackage{fullpage} \usepackage{fancybox} \usepackage{amsmath} +\usepackage{amscd} \usepackage{moreverb} \usepackage{commath} +\usepackage{algorithm2e} +\usepackage{listings} +\usepackage[standard]{ntheorem} -\title{Efficient generation of pseudo random numbers based on chaotic iterations on GPU} +% Pour mathds : les ensembles IR, IN, etc. +\usepackage{dsfont} + +% Pour avoir des intervalles d'entiers +\usepackage{stmaryrd} + +\usepackage{graphicx} +% Pour faire des sous-figures dans les figures +\usepackage{subfigure} + +\usepackage{color} + +\newtheorem{notation}{Notation} + +\newcommand{\X}{\mathcal{X}} +\newcommand{\Go}{G_{f_0}} +\newcommand{\B}{\mathds{B}} +\newcommand{\N}{\mathds{N}} +\newcommand{\BN}{\mathds{B}^\mathsf{N}} +\let\sur=\overline + +\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}} + +\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations +on GPU} \begin{document} + +\author{Jacques M. Bahi, Rapha\"{e}l Couturier, and Christophe +Guyeux\thanks{Authors in alphabetic order}} + \maketitle \begin{abstract} -This is the abstract +In this paper we present a new produce pseudo-random numbers generator (PRNG) on +graphics processing units (GPU). This PRNG is based on chaotic iterations. it +is proven to be chaotic in the Devany's formulation. We propose an efficient +implementation for GPU which succeeds to the {\it BigCrush}, the hardest +batteries of test of TestU01. Experimentations show that this PRNG can generate +about 20 billions of random numbers per second on Tesla C1060 and NVidia GTX280 +cards. + + \end{abstract} \section{Introduction} -Interet des itérations chaotiques pour générer des nombre alea\\ -Interet de générer des nombres alea sur GPU -... +Random numbers are used in many scientific applications and simulations. On +finite state machines, as computers, it is not possible to generate random +numbers but only pseudo-random numbers. In practice, a good pseudo-random number +generator (PRNG) needs to verify some features to be used by scientists. It is +important to be able to generate pseudo-random numbers efficiently, the +generation needs to be reproducible and a PRNG needs to satisfy many usual +statistical properties. Finally, from our point a view, it is essential to prove +that a PRNG is chaotic. Concerning the statistical tests, TestU01 is the +best-known public-domain statistical testing package. So we use it for all our +PRNGs, especially the {\it BigCrush} which provides the largest serie of tests. +Concerning the chaotic properties, Devaney~\cite{Devaney} proposed a common +mathematical formulation of chaotic dynamical systems. + +In a previous work~\cite{bgw09:ip} we have proposed a new familly of chaotic +PRNG based on chaotic iterations. We have proven that these PRNGs are +chaotic in the Devaney's sense. In this paper we propose a faster version which +is also proven to be chaotic. + +Although graphics processing units (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 pseudo-random +numbers inside a GPU when a scientific application runs in a GPU. That is why we +also provide an efficient PRNG for GPU respecting based on IC. Such devices +allows us to generated almost 20 billions of random numbers per second. + +In order to establish that our PRNGs are chaotic according to the Devaney's +formulation, we extend what we have proposed in~\cite{guyeux10}. Moreover, we +define a new distance to measure the disorder in the chaos and we prove some +interesting properties with this distance. + +The rest of this paper is organised as follows. In Section~\ref{section:related + works} we review some GPU implementions of PRNG. Section~\ref{section:BASIC + RECALLS} gives some basic recalls on Devanay's formation of chaos and chaotic +iterations. In Section~\ref{sec:pseudo-random} the proof of chaos of our PRNGs +is studied. Section~\ref{sec:efficient prng} presents an efficient +implementation of our chaotic PRNG on a CPU. Section~\ref{sec:efficient prng + gpu} describes the GPU implementation of our chaotic PRNG. In +Section~\ref{sec:experiments} some experimentations are presented. +Section~\ref{sec:de la relativité du désordre} describes the relativity of +disorder. In Section~\ref{sec: chaos order topology} the proof that chaotic +iterations can be described by iterations on a real interval is +established. Finally, we give a conclusion and some perspectives. + + + + +\section{Related works on GPU based PRNGs} +\label{section:related works} +In the litterature many authors have work on defining GPU based PRNGs. We do not +want to be exhaustive and we just give the most significant works from our point +of view. When authors mention the number of random numbers generated per second +we mention it. We consider that a million numbers per second corresponds to +1MSample/s and than a billion numbers per second corresponds to 1GSample/s. + +In \cite{Pang:2008:cec}, the authors define a PRNG based on cellular automata +which does not require high precision integer arithmetics nor bitwise +operations. There is no mention of statistical tests nor proof that this PRNG is +chaotic. Concerning the speed of generation, they can generate about +3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now). + +In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs +based on Lagged Fibonacci, Hybrid Taus 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} test of TestU01. There is +no mention that their PRNGs have chaos mathematical properties. + + +Authors of~\cite{conf/fpga/ThomasHL09} have studied the implementation of some +PRNGs on diferrent computing architectures: CPU, field-programmable gate array +(FPGA), GPU and massively parallel processor. This study is interesting because +it shows the performance of the same PRNGs on different architeture. For +example, the FPGA is globally the fastest architecture and it is also the +efficient one because it provides the fastest number of generated random numbers +per joule. Concerning the GPU, authors can generate betweend 11 and 16GSample/s +with a GTX 280 GPU. The drawback of this work is that those PRNGs only succeed +the {\it Crush} test which is easier than the {\it Big Crush} test. +\newline +\newline +To the best of our knowledge no GPU implementation have been proven to have chaotic properties. + +\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} + +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 $f : +\mathcal{X} \rightarrow \mathcal{X}$. + +\begin{definition} +$f$ 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} + +\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}] +$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and +topologically transitive. +\end{definition} + +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} +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 $f$. +\end{definition} + +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. + + + +\subsection{Chaotic Iterations} +\label{sec:chaotic iterations} + + +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 $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$ + +\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} + +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} + + +This new distance has been introduced to satisfy the following requirements. +\begin{itemize} +\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} +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} + +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 finally proven in \cite{bcgr11:ip} that, + + +\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} + +This result of chaos has lead us to study the possibility to build a +pseudo-random 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 $f$). 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). + +\section{Application to Pseudo-Randomness} +\label{sec:pseudo-random} +\subsection{A First Pseudo-Random 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 function $f$; +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 tests~\cite{bcgr11:ip}. + +\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: -\section{Chaotic iterations} +\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} +\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. -Présentation des itérations chaotiques +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: -\section{Efficient prng based on chaotic iterations} +\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{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. -On parle du séquentiel avec des nombres 64 bits\\ -Faire le lien avec le paragraphe précédent (je considère que la stratégie s'appelle $S^i$\\ +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 pseudo-random 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. +\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 +$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$. + + +\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} + + +Before being able to study the topological behavior of the general +chaotic iterations, we must firstly establish that: + +\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{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} + + +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} + +Let us firstly prove the following lemma. + +\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} + 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 $X'$ 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} + + + +\section{Efficient PRNG based on Chaotic Iterations} +\label{sec:efficient prng} 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 +that the strategy used contains all the bits for which the negation is +achieved out. Then in order to apply the negation on these bits we can simply +apply the xor operator between the current number and the strategy. In order to obtain the strategy we also use a classical PRNG. +Here is an example with 16-bits numbers showing how the bitwise operations +are +applied. Suppose that $x$ and the strategy $S^i$ are defined in binary mode. +Then the following table shows the result of $x$ xor $S^i$. +$$ +\begin{array}{|cc|cccccccccccccccc|} +\hline +x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\ +\hline +S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\ +\hline +x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\ +\hline + +\hline + \end{array} +$$ + + + + + +\lstset{language=C,caption={C code of the sequential chaotic iterations based +PRNG},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} + + + + + +In listing~\ref{algo:seqCIprng} a sequential version of our chaotic iterations +based PRNG is presented. The xor operator is represented by \textasciicircum. +This function uses three classical 64-bits PRNG: the \texttt{xorshift}, the +\texttt{xor128} and the \texttt{xorwow}. In the following, we call them +xor-like PRNGSs. These three PRNGs are presented in~\cite{Marsaglia2003}. As +each xor-like 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}. So to produce a random number realizes 6 xor operations +with 6 32-bits numbers produced by 3 64-bits PRNG. This version successes the +BigCrush of the TestU01 battery~\cite{LEcuyerS07}. + +\section{Efficient PRNGs based on chaotic iterations on GPU} +\label{sec:efficient prng gpu} + +In order to benefit from computing power of GPU, a program needs to define +independent blocks of threads which 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 performance is +obtained on GPU. So with algorithm \ref{algo:seqCIprng} presented in the +previous section, it is possible to build a similar program which computes PRNG +on GPU. In the CUDA~\cite{Nvid10} environment, threads have a local +identificator, called \texttt{ThreadIdx} relative to the block containing them. + + +\subsection{Naive version for GPU} + +From the CPU version, it is possible to obtain a quite similar version for GPU. +The principe consists in assigning the computation of a PRNG as in sequential to +each thread of the GPU. Of course, it is essential that the three xor-like +PRNGs used for our computation have different parameters. So we chose them +randomly with another PRNG. As the initialisation is performed by the CPU, we +have chosen to use the ISAAC PRNG~\cite{Jenkins96} to initalize all the +parameters for the GPU version of our PRNG. The implementation of the three +xor-like PRNGs is straightforward as soon as their parameters have been +allocated in the GPU memory. Each xor-like PRNGs used works with an internal +number $x$ which keeps the last generated random numbers. Other internal +variables are also used by the xor-like PRNGs. More precisely, 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 for the chaotic iterations based PRNG GPU naive version} +\label{algo:gpu_kernel} +\end{algorithm} + +Algorithm~\ref{algo:gpu_kernel} presents a naive implementation of PRNG using +GPU. According 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, i.e. the variable \texttt{n} in +algorithm~\ref{algo:gpu_kernel}. For example, 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 internals variables of xor-like +PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers} +and random number of our PRNG is equals to $100,000\times ((4+5+6)\times +2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb. + +All the tests performed to pass the BigCrush of TestU01 succeeded. Different +number of threads, called \texttt{NumThreads} in our algorithm, have been tested +upto $10$ millions. + +\begin{remark} +Algorithm~\ref{algo:gpu_kernel} has the advantage to manipulate independent +PRNGs, so this version is easily usable on a cluster of computer. The only thing +to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in +using a master node for the initialization which 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., using less than 3 xor-like PRNGs. The solution consists in computing only +one xor-like PRNG by thread, saving it into shared memory and using the results +of some other threads in the same block of threads. In order to define which +thread uses the result of which other one, we can use a permutation array which +contains the indexes of all threads and for which a permutation has been +performed. In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used. +The variable \texttt{offset} is computed using the value of +\texttt{permutation\_size}. Then we can compute \texttt{o1} and \texttt{o2} +which represent the indexes of the other threads for which the results are used +by the current thread. In the algorithm, we consider that a 64-bits xor-like +PRNG is used, that is why both 32-bits parts are used. + +This version also succeeds to the {\it BigCrush} batteries of tests. + +\begin{algorithm} + +\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs +in global memory\; +NumThreads: Number of threads\; +tab1, tab2: Arrays containing permutations of size permutation\_size\;} + +\KwOut{NewNb: array containing random numbers in global memory} +\If{threadId is concerned} { + retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory\; + offset = threadIdx\%permutation\_size\; + o1 = threadIdx-offset+tab1[offset]\; + o2 = threadIdx-offset+tab2[offset]\; + \For{i=1 to n} { + t=xor-like()\; + t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\; + shared\_mem[threadId]=t\; + x = x $\oplus$ 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{algorithm} + +\subsection{Theoretical Evaluation of the Improved Version} + +A run of Algorithm~\ref{algo:gpu_kernel2} consists in three operations having +the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative +system of Eq.~\ref{eq:generalIC}. That is, three iterations of the general chaotic +iterations are realized between two stored values of the PRNG. +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 each right +term, corresponding to terms of the strategies, can possibly be equal to any +integer of $\llbracket 1, \mathsf{N} \rrbracket$. + +Such a result is obvious for the two first lines, as for the xor-like(), all the +integers belonging into its interval of definition can occur at each iteration. +It can be easily stated for the two last lines by an immediate mathematical +induction. + +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 computer equiped with Tesla C1060 NVidia GPU card and an +Intel Xeon E5530 cadenced at 2.40 GHz for our experiments and we have used +another one equipped with a less performant CPU and a GeForce GTX 280. Both +cards have 240 cores. + +In Figure~\ref{fig:time_gpu} we compare the number of random numbers generated +per second. The xor-like prng is a xor64 described in~\cite{Marsaglia2003}. In +order to obtain the optimal performance we remove the storage of random numbers +in the GPU memory. This step is time consumming and slows down the random number +generation. Moreover, if you are interested by applications that consome random +numbers directly when they are generated, their storage is completely +useless. In this figure we can see that when the number of threads is greater +than approximately 30,000 upto 5 millions the number of random numbers generated +per second is almost constant. With the naive version, it is between 2.5 and +3GSample/s. With the optimized version, it is approximately equals to +20GSample/s. Finally we can remark that both GPU cards are quite similar. In +practice, the Tesla C1060 has more memory than the GTX 280 and this memory +should be of better quality. + \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\^\ (unsigned int)t1;\\ - x = x\^\ (unsigned int)(t2$>>$32);\\ - x = x\^\ (unsigned int)(t3$>>$32);\\ - x = x\^\ (unsigned int)t2;\\ - x = x\^\ (unsigned int)(t1$>>$32);\\ - x = x\^\ (unsigned int)t3;\\ - return x;\\ -\} -\end{minipage} -} + \includegraphics[scale=.7]{curve_time_gpu.pdf} \end{center} -\caption{sequential Chaotic Iteration PRNG} -\label{algo:seqCIprng} +\caption{Number of random numbers generated per second} +\label{fig:time_gpu} \end{figure} -In Figure~\ref{algo:seqCIprng} a sequential version of our chaotic iterations -based PRNG is presented. This version uses three classical 64-bits PRNG: the -\texttt{xorshift}, the \texttt{xor128} and the \texttt{xorwow}. These three -PRNGs are presented in~\cite{Marsaglia2003}. As each PRNG used works with -64-bits and as our PRNG works with 32-bits, the use of \texttt{(unsigned int)} -selects the 32 least significant bits whereas \texttt{(unsigned int)(t3$>>$32)} -selects the 32 bits most significants bits of the variable \texttt{t}. -\section{Efficient prng based on chaotic iterations on GPU} +In comparison, Listing~\ref{algo:seqCIprng} allows us to generate about +138MSample/s with only one core of the Xeon E5530. + + + + + +\section{The relativity of disorder} +\label{sec:de la relativité du désordre} + +In the next two sections, we investigate the impact of the choices that have +lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}. + +\subsection{Impact of the topology's finenesse} + +Let us firstly introduce the following notations. + +\begin{notation} +$\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} + + + +\begin{theorem} +\label{Th:chaos et finesse} +Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t. +$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous +both for $\tau$ and $\tau'$. + +If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then +$(\mathcal{X}_\tau,f)$ is chaotic too. +\end{theorem} + +\begin{proof} +Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$. + +Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in +\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we +can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) = +\varnothing$. Consequently, $f$ is $\tau-$transitive. + +Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for +all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a +periodic point for $f$ into $V$. + +Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood +of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$. + +But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in +\mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a +periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is +proven. +\end{proof} + +\subsection{A given system can always be claimed as chaotic} + +Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point. +Then this function is chaotic (in a certain way): + +\begin{theorem} +Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having +at least a fixed point. +Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete) +topology on $\X$. +\end{theorem} + + +\begin{proof} +$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus +\{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq +\varnothing$. +As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for +an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For +instance, $n=0$ is appropriate. + +Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V = +\mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is +regular, and the result is established. +\end{proof} + + + + +\subsection{A given system can always be claimed as non-chaotic} + +\begin{theorem} +Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$. +If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic +(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology. +\end{theorem} + +\begin{proof} +Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty}, +f\right)$ is both transitive and regular. + +Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must +contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty}, +f\right)$ is regular. Then $x$ must be a periodic point of $f$. + +Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite +because $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in +\mathcal{X}, y \notin I_x$. + +As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty +sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq +\varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x +\Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$. +\end{proof} + + + + + + +\section{Chaos on the order topology} +\label{sec: chaos order topology} +\subsection{The phase space is an interval of the real line} + +\subsubsection{Toward a topological semiconjugacy} + +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. + +Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket +1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N} +\times \B^\mathsf{N}$. + + +\begin{definition} +The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[ +0, 2^{10} \big[$ is defined by: +\begin{equation} + \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} +\end{equation} +where $\varphi\left((S,E)\right)$ is the real number: +\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 $s = 0,S^0~ S^1~ S^2~ \hdots = +\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ +\end{itemize} +\end{definition} + + + +$\varphi$ 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: + + +\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{equation} +\begin{array}{cccl} +e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\ + & x & \longmapsto & (e_0, \hdots, e_9) +\end{array} +\end{equation} +and +\begin{equation} + \begin{array}{cccc} +s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 +\rrbracket^{\mathds{N}} \\ + & x & \longmapsto & (s^k)_{k \in \mathds{N}} +\end{array} +\end{equation} +\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{definition} +$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by: +\begin{equation} +\begin{array}{cccc} +g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\ + & x & \longmapsto & g(x) +\end{array} +\end{equation} +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: + \begin{equation} +e_i' = \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} +\right. +\end{equation} +\item whose decimal part is $s(x)^1, s(x)^2, \hdots$ +\end{itemize} +\end{definition} + +\bigskip + + +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: +\begin{equation} +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}}. +\end{equation} + + +\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$} + +Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most +usual one being the Euclidian distance recalled bellow: + +\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} + +\medskip + +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: + + + +\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} + +\begin{proposition} +$D$ is a distance on $\big[ 0, 2^{10} \big[$. +\end{proposition} + +\begin{proof} +The three axioms defining a distance must be checked. +\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. +\end{itemize} +\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} + + + + +\subsubsection{The semiconjugacy} + +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 $\big[ 0, 2^\mathsf{N} +\big[$. + + + + + + +\subsection{Study of the chaotic iterations described as a real function} + + +\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{figure}[t] +\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}} +\end{center} +\caption{ICs on small intervals.} +\label{fig:ICs2} +\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} + + +\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} + +\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} + +The sequential characterization of the continuity concludes the demonstration. +\end{proof} + + + +A contrario: + +\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. +\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. +\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: + +\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 +$\tau_\Delta$. +\end{itemize} + + +\subsection{Chaos of the chaotic iterations on $\mathds{R}$} +\label{chpt:Chaos des itérations chaotiques sur R} + + + +\subsubsection{Chaos according to Devaney} + +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: +\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 $\mathds{R}$. +\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. + + + -On parle du passage du sequentiel au GPU -\section{Experiments} -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{Lyapunov} \section{Conclusion} \bibliographystyle{plain}