X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/5e488f8546b2d4aaeefee70e1adfa87435312a6f..3abda4f59d76446238e6b891bc14e1ac7b44c34a:/prng_gpu.tex?ds=sidebyside diff --git a/prng_gpu.tex b/prng_gpu.tex index 25214fd..f357476 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -192,12 +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 @@ -205,8 +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 @@ -726,15 +725,7 @@ 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} + The general chaotic iterations defined in Equation~\ref{eq:generalIC} satisfy the Devaney's property of chaos. \end{theorem} @@ -998,10 +989,10 @@ satisfy %%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} @@ -1325,7 +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 @@ -1647,13 +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 @@ -1662,7 +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 $uv$. @@ -1775,9 +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 @@ -2019,7 +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.