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+\usepackage{cite}
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% Pour mathds : les ensembles IR, IN, etc.
\IEEEcompsoctitleabstractindextext{
\begin{abstract}
In this paper we present a new pseudorandom number generator (PRNG) on
-graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations. It
-is firstly proven to be chaotic according to the Devaney's formulation. We thus propose an efficient
+graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations and
+it is thus chaotic according to the Devaney's formulation. We propose an efficient
implementation for GPU that successfully passes the {\it BigCrush} tests, deemed to be the hardest
battery of tests in TestU01. Experiments show that this PRNG can generate
about 20 billion of random numbers per second on Tesla C1060 and NVidia GTX280
and on an iteration process called ``chaotic
iterations'' on which the post-treatment is based.
The proposed PRNG and its proof of chaos are given in Section~\ref{sec:pseudorandom}.
-
-Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
-improvement related to the chaotic iteration based post-treatment, for
-our previously released PRNGs and a new efficient
+Section~\ref{sec:efficient PRNG} %{The generation of pseudorandom sequence} %illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and
+ contains a new efficient
implementation on CPU.
-
Section~\ref{sec:efficient PRNG
gpu} describes and evaluates theoretically the GPU implementation.
Such generators are experimented in
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
+\ref{Def:chaotic iterations}. The question to determine whether the
use of more general chaotic iterations to generate pseudorandom numbers
-faster, does not deflate their topological chaos properties.
+faster, does not deflate their topological chaos properties, has been
+investigated in Annex~\ref{A-deuxième def}, leading to the following result.
+
+ \begin{theorem}
+ \label{t:chaos des general}
+ The general chaotic iterations defined in Equation~\ref{eq:generalIC}
+satisfy
+ the Devaney's property of chaos.
+ \end{theorem}
%%RAF proof en supplementary, j'ai mis le theorem.
% A vérifier
- \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
-\label{deuxième def}
-The proof is given in Section~\ref{A-deuxième def} of the annex document.
+% \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+%\label{deuxième def}
+%The proof is given in Section~\ref{A-deuxième def} of the annex document.
%% \label{deuxième def}
%% Let us consider the discrete dynamical systems in chaotic iterations having
%% the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in
%%RAF : mis en supplementary
-\section{Statistical Improvements Using Chaotic Iterations}
-\label{The generation of pseudorandom sequence}
-The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
-
+%\section{Statistical Improvements Using Chaotic Iterations}
+%\label{The generation of pseudorandom sequence}
+%The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
+The reasons to desire chaos to achieve randomness are given in Annex~\ref{A-The generation of pseudorandom sequence}.
%% \label{The generation of pseudorandom sequence}
%% raise ambiguity.
-\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
+\section{First Efficient Implementation of a PRNG based on Chaotic Iterations}
\label{sec:efficient PRNG}
%
%Based on the proof presented in the previous section, it is now possible to
