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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
than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
statistical behavior). Experiments are also provided using BBS as the initial
random generator. The generation speed is significantly weaker.
-Note also that an original qualitative comparison between topological chaotic
-properties and statistical test is also proposed.
+%Note also that an original qualitative comparison between topological chaotic
+%properties and statistical test is also proposed.
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
We show in Section~\ref{sec:security analysis} that, if the inputted
generator is cryptographically secure, then it is the case too for the
generator provided by the post-treatment.
-A practical
-security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
+%A practical
+%security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
Such a proof leads to the proposition of a cryptographically secure and
chaotic generator on GPU based on the famous Blum Blum Shub
in Section~\ref{sec:CSGPU} and to an improvement of the
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
This section is dedicated to the security analysis of the
- proposed PRNGs, both from a theoretical and from a practical point of view.
+ proposed PRNGs.%, both from a theoretical and from a practical point of view.
-\subsection{Theoretical Proof of Security}
+%\subsection{Theoretical Proof of Security}
\label{sec:security analysis}
The standard definition
- of {\it indistinguishability} used is the classical one as defined for
+ of {\it indistinguishability} used here is the classical one as defined for
instance in~\cite[chapter~3]{Goldreich}.
This property shows that predicting the future results of the PRNG
cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this
be broken in practice. But it also means that if the keys/seeds are large
enough, the system is secured.
As a complement, an example of a concrete practical evaluation of security
-is outlined in the next subsection.
+is outlined in Annex~\ref{A-sec:Practicak evaluation}.
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
-\subsection{Practical Security Evaluation}
-\label{sec:Practicak evaluation}
-This subsection is given in Section~\ref{A-sec:Practicak evaluation} of the annex document.
+%\subsection{Practical Security Evaluation}
+%\label{sec:Practicak evaluation}
+%This subsection is given in Section
+A example of a practical security evaluation is outlined in
+Annex~\ref{A-sec:Practicak evaluation}.
%%RAF mis en annexe
proposed parameters, or if it is only a very fast
and statistically perfect generator on GPU, its
$(T,\varepsilon)-$security must be determined, and
-a formulation similar to Eq.\eqref{mesureConcrete}
+a formulation similar to Annex~\ref{A-sec:Practicak evaluation} %.Eq.\eqref{mesureConcrete}
must be established. Authors
hope to achieve this difficult task in a future
work.
