\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
-\title{Efficient Generation of Pseudo-Random Bumbers 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}
+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. Devaney~\cite{Devaney} proposed a common mathematical
+formulation of chaotic dynamical systems. Concerning the statistical tests,
+TestU01the is the best-known public-domain statistical testing packages. So we
+use it for all our PRNGs, especially the {\it BigCrush} which is based on the largest
+serie of tests.
+
+In a previous work~\cite{bgw09:ip} we have proposed a new familly of chaotic
+PRNG based on chaotic iterations (IC). In this paper we propose a faster
+version which is also proven to be chaotic with the Devaney formulation.
+
+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.
+
+
+
+
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 ?}
-...
+\section{Related works on GPU based PRNGs}
+
+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.
+
+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 3200000
+random numbers per seconds 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.
+
\section{Basic Recalls}
\label{section:BASIC RECALLS}
This section is devoted to basic definitions and terminologies in the fields of
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
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:
We can now prove the Theorem~\ref{t:chaos des general}...
\begin{proof}[Theorem~\ref{t:chaos des general}]
- On the one hand, strong transitivity implies transitivity. On the other hand,
-the regularity is exactly Lemma~\ref{strongTrans} with $Y=X$. As the sensitivity
-to the initial condition is implied by these two properties, we thus have
-the theorem.
+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}
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$>>$32)} selects the 32
-most significants bits of the variable \texttt{t}. So to produce a random
-number realizes 6 xor operations with 6 32-bits numbers produced by 3 64-bits
-PRNG. This version successes the BigCrush of the TestU01 battery [P. L’ecuyer
- and R. Simard. Testu01].
+based PRNG is presented. The xor operator is represented by \textasciicircum.
+This function uses three classical 64-bits PRNG: the \texttt{xorshift}, the
+\texttt{xor128} and the \texttt{xorwow}. In the following, we call them
+xor-like PRNGSs. These three PRNGs are presented in~\cite{Marsaglia2003}. As
+each xor-like PRNG used works with 64-bits and as our PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$32)} selects the 32 most significants bits of the
+variable \texttt{t}. So to produce a random number realizes 6 xor operations
+with 6 32-bits numbers produced by 3 64-bits PRNG. This version successes the
+BigCrush of the TestU01 battery~\cite{LEcuyerS07}.
\section{Efficient prng based on chaotic iterations on GPU}
As GPU cards using CUDA have shared memory between threads of the same block, it
is possible to use this feature in order to simplify the previous algorithm,
-i.e. using less than 3 xor-like PRNGs. The solution consists in computing only
+i.e., using less than 3 xor-like PRNGs. The solution consists in computing only
one xor-like PRNG by thread, saving it into shared memory and using the results
of some other threads in the same block of threads. In order to define which
thread uses the result of which other one, we can use a permutation array which
\label{algo:gpu_kernel2}
\end{algorithm}
-
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, four iterations of the general chaotic
+iterations are realized between two stored values of the PRNG.
+To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+we must guarantee that this dynamical system iterates on the space
+$\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
+Different experiments have been performed in order to measure the generation
speed.
\begin{figure}[t]
\begin{center}
\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.
