From: cguyeux Date: Mon, 3 Sep 2012 07:09:09 +0000 (+0200) Subject: avancées dans la réécriture X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/commitdiff_plain/9879779d913285ee14baad568f69be401dfd0fb3?ds=sidebyside avancées dans la réécriture --- diff --git a/prng_gpu.tex b/prng_gpu.tex index c7853b2..0c9f9c7 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -90,7 +90,13 @@ On the other side, speed is not the main requirement in cryptography: the great need is to define \emph{secure} generators able to withstand malicious attacks. Roughly speaking, an attacker should not be able in practice to make the distinction between numbers obtained with the secure generator and a true random -sequence. +sequence. \begin{color}{red} Or, in an equivalent formulation, he or she should not be +able (in practice) to predict the next bit of the generator, having the knowledge of all the +binary digits that have been already released. ``Being able in practice'' refers here +to the possibility to achieve this attack in polynomial time, and to the exponential growth +of the difficulty of this challenge when the size of the parameters of the PRNG increases. +\end{color} + Finally, a small part of the community working in this domain focuses on a third requirement, that is to define chaotic generators. The main idea is to take benefits from a chaotic dynamical system to obtain a @@ -124,18 +130,18 @@ statistical perfection refers to the ability to pass the whole {\it BigCrush} battery of tests, which is widely considered as the most stringent statistical evaluation of a sequence claimed as random. This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}. -Chaos, for its part, refers to the well-established definition of a -chaotic dynamical system proposed by Devaney~\cite{Devaney}. \begin{color}{red} More precisely, each time we performed a test on a PRNG, we ran it -twice in order to observe if all p-values are inside [0.01, 0.99]. In -fact, we observed that few p-values (less than ten) are sometimes +twice in order to observe if all $p-$values are inside [0.01, 0.99]. In +fact, we observed that few $p-$values (less than ten) are sometimes outside this interval but inside [0.001, 0.999], so that is why a second run allows us to confirm that the values outside are not for the same test. With this approach all our PRNGs pass the {\it - BigCrush} successfully and all p-values are at least once inside + BigCrush} successfully and all $p-$values are at least once inside [0.01, 0.99]. \end{color} +Chaos, for its part, refers to the well-established definition of a +chaotic dynamical system proposed by Devaney~\cite{Devaney}. In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave as a chaotic dynamical system. Such a post-treatment leads to a new category of @@ -166,8 +172,13 @@ 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{sec:efficient PRNG} presents an efficient -implementation of this chaotic PRNG on a CPU, whereas Section~\ref{sec:efficient PRNG +\begin{color}{red} +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 +implementation on CPU. +\end{color} + Section~\ref{sec:efficient PRNG gpu} describes and evaluates theoretically the GPU implementation. Such generators are experimented in Section~\ref{sec:experiments}. @@ -176,7 +187,8 @@ generator is cryptographically secure, then it is the case too for the generator provided by the post-treatment. 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 +in Section~\ref{sec:CSGPU}, \begin{color}{red} to a practical +security evaluation in Section~\ref{sec:Practicak evaluation}, \end{color} and to an improvement of the Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}. This research work ends by a conclusion section, in which the contribution is summarized and intended future work is presented. @@ -184,7 +196,7 @@ summarized and intended future work is presented. -\section{Related works on GPU based PRNGs} +\section{Related work on GPU based PRNGs} \label{section:related works} Numerous research works on defining GPU based PRNGs have already been proposed in the @@ -584,8 +596,8 @@ $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}. The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}. -This function is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$ -(the reader is referred to~\cite{bg10:ip} for more information). +This function must be chosen such that the outputs of the resulted PRNG is uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this +goal (other candidates and more information can be found in ~\cite{bg10:ip}). \begin{equation} \label{Formula} @@ -651,7 +663,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n, \end{array} \right. -\label{equation Oplus0} +\label{equation Oplus} \end{equation} where $\oplus$ is for the bitwise exclusive or between two integers. This rewriting can be understood as follows. The $n-$th term $S^n$ of the @@ -661,7 +673,7 @@ as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th component of this state (a binary digit) changes if and only if the $k-$th digit in the binary decomposition of $S^n$ is 1. -The single basic component presented in Eq.~\ref{equation Oplus0} is of +The single basic component presented in Eq.~\ref{equation Oplus} is of ordinary use as a good elementary brick in various PRNGs. It corresponds to the following discrete dynamical system in chaotic iterations: @@ -683,7 +695,7 @@ we select a subset of components to change. Obviously, replacing the previous CI PRNG Algorithms by -Equation~\ref{equation Oplus0}, which is possible when the iteration function is +Equation~\ref{equation Oplus}, which is possible when the iteration function is the vectorial negation, leads to a speed improvement (the resulting generator will be referred as ``Xor CI PRNG'' in what follows). @@ -1221,7 +1233,7 @@ raise ambiguity. -\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG} +\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}} \begin{small} \begin{lstlisting} @@ -1731,6 +1743,7 @@ secure. \begin{color}{red} \subsection{Practical Security Evaluation} +\label{sec:Practicak evaluation} Suppose now that the PRNG will work during $M=100$ time units, and that during this period,