\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
-\title{Efficient generation of pseudo random numbers based on chaotic iterations
+\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
on GPU}
\begin{document}
\maketitle
\begin{abstract}
-This is the abstract
+
\end{abstract}
\section{Introduction}
-Interet des itérations chaotiques pour générer des nombre alea\\
-Interet de générer des nombres alea sur GPU
-\alert{RC, un petit state-of-the-art sur les PRNGs 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 (IC). 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}
+\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 $V$. $f^{k}=f\circ ...\circ f$
-denotes the $k^{th}$ composition of a function $f$. Finally, the following
+is for the $k^{th}$ composition of a function $f$. Finally, the following
notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
\end{definition}
-\begin{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}
-\subsection{Chaotic iterations}
+\subsection{Chaotic Iterations}
\label{sec:chaotic iterations}
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 $\mathbb{S}.$
+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 $S\in \mathbb{S}$ be a strategy. The so-called
+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}
$\left(f(x^{k})\right)_{S^{n}}$, 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, recalled above.
+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
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 $\sigma
-(S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in
-\mathds{N}}\in \mathbb{S}$ and $i$ is the \emph{initial function}
-$i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket
-1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in
-(\ref{sec:chaotic iterations}) can be described by the following iterations:
+(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}
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:
\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. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a
+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 $\mathds{B}^n$ to itself. Then $G_{f}$ is continuous in
+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 \cite{guyeux10}. To obtain a characterization, we have secondly
+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 $\mathds{B}^n$ to itself. The
+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
-$\mathds{B}^n$; for all $x\in\mathds{B}^n$ and $i\in \llbracket1;n\rrbracket$,
+$\mathds{B}^\mathsf{N}$; 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
\begin{theorem}
\label{Th:Caractérisation des IC chaotiques}
-Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic (according to Devaney)
+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 $\llbracket 1 ; n \rrbracket^{\mathds{N}}
-\times \mathds{B}^n$, is build from Boolean networks $f : \mathds{B}^n
-\rightarrow \mathds{B}^n$, we can preserve the theoretical properties on $G_f$
+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
-$\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ; n
+$\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
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}
+\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
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
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 more
-fastly, does not deflate their topological chaos properties.
-
-\subsection{Proofs of chaos of the general formulation of the chaotic iterations}
+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:
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
Let us introduce the following function:
\begin{equation}
\begin{array}{cccc}
- \delta: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
+ \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}
Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
\begin{equation}
\begin{array}{lrll}
-F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
+F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \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},%
+& (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}%
-\noindent where + and . are the Boolean addition and product operations.
+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} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
+\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 $\mathcal{X}$:
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 $\sigma
-(S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in
-\mathds{N}}\in \mathbb{S}$ and $i$ is the \emph{initial function}
-$i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket
-1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in
-(\ref{sec:chaotic iterations}) can be described by the following iterations:
+(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}
\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.
+Another time, a shift function appears as a component of these general chaotic
+iterations.
-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:
+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})}, \\
+}\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-\check{S}^k|}{10^{k}}}.%
+\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
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$>>$32)} selects the 32
-most significants bits of the variable \texttt{t}. So to produce a random
-number realizes 6 xor operations with 6 32-bits numbers produced by 3 64-bits
-PRNG. This version successes the BigCrush of the TestU01 battery [P. L’ecuyer
- and R. Simard. Testu01].
-
-\section{Efficient prng based on chaotic iterations on GPU}
+based PRNG is presented. The xor operator is represented by \textasciicircum.
+This function uses three classical 64-bits PRNG: the \texttt{xorshift}, the
+\texttt{xor128} and the \texttt{xorwow}. In the following, we call them
+xor-like PRNGSs. These three PRNGs are presented in~\cite{Marsaglia2003}. As
+each xor-like PRNG used works with 64-bits and as our PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$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,
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 [ref] environment, threads have a local identificator,
-called \texttt{ThreadIdx} relative to the block containing them.
+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}
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 [ref] 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.
+have chosen to use the ISAAC PRNG~\ref{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}
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
+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
\label{algo:gpu_kernel2}
\end{algorithm}
-
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, four 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}
-
-Differents experiments have been performed in order to measure the generation
-speed.
-\begin{figure}[t]
+\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.
+
+In Figure~\ref{fig:time_gpu} we compare the number of random numbers generated
+per second. In order to obtain the optimal number 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 number 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 almost equals to
+20GSample/s.
+
+\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.7]{curve_time_gpu.pdf}
\end{center}
\caption{Number of random numbers generated per second}
-\label{fig:time_naive_gpu}
+\label{fig:time_gpu}
\end{figure}
\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.
\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}
