X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/a17d0a6c41e0afab24e5a0c8c806533db3ad753a..9fb4290afcf7de31edb7dda484ba7a2fedadaafb:/prng_gpu.tex?ds=sidebyside diff --git a/prng_gpu.tex b/prng_gpu.tex index 11a1d56..bc06797 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -14,6 +14,7 @@ \usepackage{algorithmic} \usepackage{slashbox} \usepackage{ctable} +\usepackage{cite} \usepackage{tabularx} \usepackage{multirow} % Pour mathds : les ensembles IR, IN, etc. @@ -54,8 +55,8 @@ Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}} \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 @@ -179,8 +180,8 @@ Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster 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. @@ -191,12 +192,11 @@ The remainder of this paper is organized as follows. In Section~\ref{section:re 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 @@ -204,8 +204,8 @@ Section~\ref{sec:experiments}. 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 @@ -718,17 +718,25 @@ in what follows). 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 @@ -981,10 +989,10 @@ The proof is given in Section~\ref{A-deuxième def} of the annex document. %%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} @@ -1308,7 +1316,7 @@ The content is this section is given in Section~\ref{A-The generation of pseudor %% 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 @@ -1630,13 +1638,13 @@ as it is shown in the next sections. 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 @@ -1645,7 +1653,7 @@ The standard definition 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 $uv$. @@ -1758,9 +1766,11 @@ proving that $H$ is not secure, which is a contradiction. -\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 @@ -2002,7 +2012,7 @@ on GPU can be useful in security context with the 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.