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.
+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
+The rest of this paper is organised as follows. In Section~\ref{section:related
+ works} we review some GPU implementions of PRNG. Section~\ref{sec:chaotic
+ iterations} 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.
-Interet des itérations chaotiques pour générer des nombre alea\\
-Interet de générer des nombres alea sur GPU
\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.
+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 3200000
-random numbers per seconds on a GeForce 7800 GTX GPU (which is quite old now).
+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
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}
\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
\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
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}
+\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,
Devaney's formulation of a chaotic behavior.
\section{Experiments}
+\label{sec:experiments}
Different experiments have been performed in order to measure the generation
speed.
\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}
