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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
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 by
+ \begin{equation}
+ x_i^n=\left\{
+ \begin{array}{ll}
+ 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}
+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
