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.
+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). In this paper we propose a faster
-version which is also proven to be chaotic with the Devaney formulation.
+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.
+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.
-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
-PRNGs for Langevin simulations of biomolecules fully implemented on GPU.
+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}
\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,
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}
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 succeed to the BigCrush batteries of tests.
+This version also succeeds to the {\it BigCrush} batteries of tests.
\begin{algorithm}
Devaney's formulation of a chaotic behavior.
\section{Experiments}
-
-Different 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}
-First of all we have compared the time to generate X random numbers with both
-the CPU version and the GPU version.
+In comparison, Listing~\ref{algo:seqCIprng} allows us to generate about
+138MSample/s with only one core of the Xeon E5530.
+
-Faire une courbe du nombre de random en fonction du nombre de threads,
-éventuellement en fonction du nombres de threads par bloc.
\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}
