X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/fc073e86031fc91a73a2f1c436f91a697cb98668..baf2ff7c5bd7a44b68600f37eb61b76e67a8be17:/prng_gpu.tex?ds=inline diff --git a/prng_gpu.tex b/prng_gpu.tex index 085ce1e..4880738 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -40,14 +40,25 @@ \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}} +\begin{document} +\begin{frontmatter} +\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU} + + +\author{Jacques M. Bahi} +\ead{jacques.bahi@univ-fcomte.fr} +\author{ Rapha\"{e}l Couturier \corref{cor1}} +\ead{raphael.couturier@univ-fcomte.fr} +\cortext[cor1]{Corresponding author} +\author{ Christophe Guyeux} +\ead{christophe.guyeux@univ-fcomte.fr} +\author{ Pierre-Cyrille Héam } +\ead{pierre-cyrille.heam@univ-fcomte.fr} + +\address{FEMTO-ST Institute, UMR 6174 CNRS,\\ University of Franche Comte, Belfort, France\\ Authors in alphabetic order} -\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU} -\begin{document} -\author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe -Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}} - %\IEEEcompsoctitleabstractindextext{ \begin{abstract} @@ -65,8 +76,13 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin \end{abstract} %} +\begin{keyword} + pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos + +\end{keyword} +\end{frontmatter} -\maketitle +%\maketitle %\IEEEdisplaynotcompsoctitleabstractindextext %\IEEEpeerreviewmaketitle @@ -92,7 +108,7 @@ 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. However, in an equivalent formulation, he or she should not be +sequence. 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 @@ -141,7 +157,7 @@ the same test. With this approach all our PRNGs pass the {\it BigCrush} successfully and all $p-$values are at least once inside [0.01, 0.99]. Chaos, for its part, refers to the well-established definition of a -chaotic dynamical system proposed by Devaney~\cite{Devaney}. +chaotic dynamical system defined 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 @@ -177,8 +193,8 @@ 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 tests is also proposed. @@ -189,14 +205,14 @@ 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 -implementation on CPU. - - Section~\ref{sec:efficient PRNG - gpu} describes and evaluates theoretically the GPU implementation. +%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. + Section~\ref{sec:efficient PRNG} %{sec:efficient PRNG +% gpu} + describes and evaluates theoretically new effective versions of +our pseudorandom generators, in particular with a GPU implementation. Such generators are experimented in Section~\ref{sec:experiments}. We show in Section~\ref{sec:security analysis} that, if the inputted @@ -520,7 +536,7 @@ two PRNGs as inputs. These two generators are mixed with chaotic iterations, leading thus to a new PRNG that should improve the statistical properties of each generator taken alone. -Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as present input. +Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as input present. @@ -662,6 +678,22 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\ \end{algorithmic} \end{algorithm} + +We have shown in~\cite{bfg12a:ip} that the use of chaotic iterations +implies an improvement of the statistical properties for all the +inputted defective generators we have investigated. +For instance, when considering the TestU01 battery with its 588 tests, we obtained 261 +failures for a PRNG based on the logistic map alone, and +this number of failures falls below 138 in the Old CI(Logistic,Logistic) generator. +In the XORshift case (146 failures when considering it alone), the results are more amazing, +as the chaotic iterations post-treatment makes it fails only 8 tests. +Further investigations have been systematically realized in \cite{bfg12a:ip} +using a large set of inputted defective PRNGs, the three most used batteries of +tests (DieHARD, NIST, and TestU01), and for all the versions of generators we have proposed. +In all situations, an obvious improvement of the statistical behavior has +been obtained, reinforcing the impression that chaos leads to statistical +enhancement~\cite{bfg12a:ip}. + \subsection{Improving the Speed of the Former Generator} Instead of updating only one cell at each iteration, we now propose to choose a @@ -1293,9 +1325,9 @@ have $d((S,E),(\tilde S,E))<\epsilon$. \section{Toward Efficiency and Improvement for CI PRNG} +\label{sec:efficient PRNG} \subsection{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 %improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. @@ -1770,14 +1802,7 @@ Let $\varepsilon > 0$. $\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom generator $G$ if -\begin{flushleft} -$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$ -\end{flushleft} - -\begin{flushright} -$ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$ -\end{flushright} - +$$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right. - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$$ \noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation ``$\in_R$'' indicates the process of selecting an element at random and uniformly over the corresponding set. @@ -2052,10 +2077,10 @@ To encrypt his message, Bob will compute c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right. \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right) \end{equation*} -instead of $$\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$$. +instead of $$\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right).$$ The same decryption stage as in Blum-Goldwasser leads to the sequence -$$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$$. +$$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right).$$ Thus, with a simple use of $S^0$, Alice can obtain the plaintext. By doing so, the proposed generator is used in place of BBS, leading to the inheritance of all the properties presented in this paper.