X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/ef5711321ecf848ad064bef33993b326d804152c..3010272fc200ffae4e9223ba48c5f3caf05a4256:/prng_gpu.tex diff --git a/prng_gpu.tex b/prng_gpu.tex index de8f464..55fc756 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1,4 +1,5 @@ -\documentclass{article} +%\documentclass{article} +\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{fullpage} @@ -38,17 +39,17 @@ \begin{document} \author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe -Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}} +Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}} -\maketitle +\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 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 billions of random numbers per second on Tesla C1060 and NVidia GTX280 +about 20 billion of random numbers per second on Tesla C1060 and NVidia GTX280 cards. It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically secure. @@ -56,10 +57,17 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin \end{abstract} +} + +\maketitle + +\IEEEdisplaynotcompsoctitleabstractindextext +\IEEEpeerreviewmaketitle + \section{Introduction} -Randomness is of importance in many fields as scientific simulations or cryptography. +Randomness is of importance in many fields such as scientific simulations or cryptography. ``Random numbers'' can mainly be generated either by a deterministic and reproducible algorithm called a pseudorandom number generator (PRNG), or by a physical non-deterministic process having all the characteristics of a random noise, called a truly random number @@ -67,18 +75,18 @@ generator (TRNG). In this paper, we focus on reproducible generators, useful for instance in Monte-Carlo based simulators or in several cryptographic schemes. These domains need PRNGs that are statistically irreproachable. -On some fields as in numerical simulations, speed is a strong requirement +In some fields such as in numerical simulations, speed is a strong requirement that is usually attained by using parallel architectures. In that case, -a recurrent problem is that a deflate of the statistical qualities is often +a recurrent problem is that a deflation of the statistical qualities is often reported, when the parallelization of a good PRNG is realized. This is why ad-hoc PRNGs for each possible architecture must be found to achieve both speed and randomness. On the other side, speed is not the main requirement in cryptography: the great -need is to define \emph{secure} generators being able to withstand malicious +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. -Finally, a small part of the community working in this domain focus on a +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 generator that is unpredictable, disordered, sensible to its seed, or in other word chaotic. @@ -95,7 +103,7 @@ This is why the use of chaos for PRNG still remains marginal and disputable. The authors' opinion is that topological properties of disorder, as they are properly defined in the mathematical theory of chaos, can reinforce the quality of a PRNG. But they are not substitutable for security or statistical perfection. -Indeed, to the authors' point of view, such properties can be useful in the two following situations. On the +Indeed, to the authors' mind, such properties can be useful in the two following situations. On the one hand, a post-treatment based on a chaotic dynamical system can be applied to a PRNG statistically deflective, in order to improve its statistical properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}. @@ -110,7 +118,7 @@ Let us finish this paragraph by noticing that, in this paper, 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 into the well-known TestU01 package~\cite{LEcuyerS07}. +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}. @@ -131,11 +139,11 @@ applications. Therefore, it is important to be able to generate pseudorandom numbers inside a GPU when a scientific application runs in it. This remark motivates our proposal of a chaotic and statistically perfect PRNG for GPU. Such device -allows us to generated almost 20 billions of pseudorandom numbers per second. +allows us to generate almost 20 billion of pseudorandom numbers per second. Furthermore, we show that the proposed post-treatment preserves the cryptographical security of the inputted PRNG, when this last has such a property. -Last, but not least, we propose a rewritten of the Blum-Goldwasser asymmetric +Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric key encryption protocol by using the proposed method. The remainder of this paper is organized as follows. In Section~\ref{section:related @@ -165,8 +173,8 @@ summarized and intended future work is presented. \section{Related works on GPU based PRNGs} \label{section:related works} -Numerous research works on defining GPU based PRNGs have yet been proposed in the -literature, so that completeness is impossible. +Numerous research works on defining GPU based PRNGs have already been proposed in the +literature, so that exhaustivity is impossible. This is why authors of this document only give reference to the most significant attempts in this domain, from their subjective point of view. The quantity of pseudorandom numbers generated per second is mentioned here @@ -184,7 +192,7 @@ chaos or cryptography in this document. In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs based on Lagged Fibonacci or Hybrid Taus. They have used these PRNGs for Langevin simulations of biomolecules fully implemented on -GPU. Performance of the GPU versions are far better than those obtained with a +GPU. Performances of the GPU versions are far better than those obtained with a CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. However the evaluations of the proposed PRNGs are only statistical ones. @@ -200,7 +208,7 @@ However, we notice that authors can ``only'' generate between 11 and 16GSamples/ with a GTX 280 GPU, which should be compared with the results presented in this document. We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only -able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one. +able to pass the {\it Crush} battery, which is far easier than the {\it Big Crush} one. Lastly, Cuda has developed a library for the generation of pseudorandom numbers called Curand~\cite{curand11}. Several PRNGs are implemented, among @@ -210,13 +218,16 @@ their fastest version provides 15GSamples/s on the new Fermi C2050 card. But their PRNGs cannot pass the whole TestU01 battery (only one test is failed). \newline \newline -We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded. +We can finally remark that, to the best of our knowledge, no GPU implementation has been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered. \section{Basic Recalls} \label{section:BASIC RECALLS} This section is devoted to basic definitions and terminologies in the fields of -topological chaos and chaotic iterations. +topological chaos and chaotic iterations. We assume the reader is familiar +with basic notions on topology (see for instance~\cite{Devaney}). + + \subsection{Devaney's Chaotic Dynamical Systems} In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ @@ -229,7 +240,7 @@ Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : \mathcal{X} \rightarrow \mathcal{X}$. \begin{definition} -$f$ is said to be \emph{topologically transitive} if, for any pair of open sets +The function $f$ is said to be \emph{topologically transitive} if, for any pair of open sets $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq \varnothing$. \end{definition} @@ -248,7 +259,7 @@ necessarily the same period). \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}] -$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and +The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and topologically transitive. \end{definition} @@ -256,12 +267,12 @@ The chaos property is strongly linked to the notion of ``sensitivity'', defined on a metric space $(\mathcal{X},d)$ by: \begin{definition} -\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions} +\label{sensitivity} The function $f$ has \emph{sensitive dependence on initial conditions} if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that $d\left(f^{n}(x), f^{n}(y)\right) >\delta $. -$\delta$ is called the \emph{constant of sensitivity} of $f$. +The constant $\delta$ is called the \emph{constant of sensitivity} of $f$. \end{definition} Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is @@ -321,11 +332,13 @@ are continuous. For further explanations, see, e.g., \cite{guyeux10}. Let $\delta $ be the \emph{discrete Boolean metric}, $\delta (x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function: +%%RAPH : ici j'ai coupé la dernière ligne en 2, c'est moche mais bon \begin{equation} \begin{array}{lrll} F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} & \longrightarrow & \mathds{B}^{\mathsf{N}} \\ -& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta +& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ \right.\\ +& & & \left. f(E)_{k}.\overline{\delta (k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% \end{array}% \end{equation}% @@ -384,9 +397,9 @@ their distance should increase too. \item In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a -while. Indeed, the two dynamical systems start with the same initial condition, -use the same update function, and as strategies are the same for a while, then -components that are updated are the same too. +while. Indeed, both dynamical systems start with the same initial condition, +use the same update function, and as strategies are the same for a while, furthermore +updated components are the same as well. \end{itemize} The distance presented above follows these recommendations. Indeed, if the floor value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ @@ -395,7 +408,7 @@ measure of the differences between strategies $S$ and $\check{S}$. More precisely, this floating part is less than $10^{-k}$ if and only if the first $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the $k^{th}$ terms of the two strategies are different. -The impact of this choice for a distance will be investigate at the end of the document. +The impact of this choice for a distance will be investigated at the end of the document. Finally, it has been established in \cite{guyeux10} that, @@ -445,15 +458,15 @@ Finally, we have established in \cite{bcgr11:ip} that, \end{theorem} -These results of chaos and uniform distribution have lead us to study the possibility to build a +These results of chaos and uniform distribution have led us to study the possibility of building a pseudorandom number generator (PRNG) based on the chaotic iterations. As $G_f$, defined on the domain $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}} -\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N} +\times \mathds{B}^\mathsf{N}$, is built from Boolean networks $f : \mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$ during implementations (due to the discrete nature of $f$). Indeed, it is as if $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG). -Let us finally remark that the vectorial negation satisfies the hypotheses of the two theorems above. +Let us finally remark that the vectorial negation satisfies the hypotheses of both theorems above. \section{Application to Pseudorandomness} \label{sec:pseudorandom} @@ -467,8 +480,9 @@ generator taken alone. Furthermore, our generator possesses various chaos properties that none of the generators used as input present. + \begin{algorithm}[h!] -%\begin{scriptsize} +\begin{small} \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)} \KwOut{a configuration $x$ ($n$ bits)} @@ -480,12 +494,16 @@ $s\leftarrow{\textit{XORshift}(n)}$\; $x\leftarrow{F_f(s,x)}$\; } return $x$\; -%\end{scriptsize} +\end{small} \caption{PRNG with chaotic functions} \label{CI Algorithm} \end{algorithm} + + + \begin{algorithm}[h!] +\begin{small} \KwIn{the internal configuration $z$ (a 32-bit word)} \KwOut{$y$ (a 32-bit word)} $z\leftarrow{z\oplus{(z\ll13)}}$\; @@ -493,7 +511,7 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\; $z\leftarrow{z\oplus{(z\ll5)}}$\; $y\leftarrow{z}$\; return $y$\; -\medskip +\end{small} \caption{An arbitrary round of \textit{XORshift} algorithm} \label{XORshift} \end{algorithm} @@ -507,7 +525,7 @@ It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractéris an integer $b$, ensuring that the number of executed iterations is at least $b$ and at most $2b+1$; and an initial configuration $x^0$. It returns the new generated configuration $x$. Internally, it embeds two -\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that returns integers +\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that return integers uniformly distributed into $\llbracket 1 ; k \rrbracket$. \textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia, @@ -536,7 +554,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \label{equation Oplus} \end{equation} where $\oplus$ is for the bitwise exclusive or between two integers. -This rewritten can be understood as follows. The $n-$th term $S^n$ of the +This rewriting can be understood as follows. The $n-$th term $S^n$ of the sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents the list of cells to update in the state $x^n$ of the system (represented as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th @@ -560,13 +578,12 @@ where $f$ is the vectorial negation and $\forall n \in \mathds{N}$, $\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that $k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary decomposition of $S^n$ is 1. Such chaotic iterations are more general -than the ones presented in Definition \ref{Def:chaotic iterations} for -the fact that, instead of updating only one term at each iteration, +than the ones presented in Definition \ref{Def:chaotic iterations} because, instead of updating only one term at each iteration, we select a subset of components to change. Obviously, replacing Algorithm~\ref{CI Algorithm} by -Equation~\ref{equation Oplus}, 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. However, proofs of chaos obtained in~\cite{bg10:ij} have been established only for chaotic iterations of the form presented in Definition @@ -606,12 +623,13 @@ Let us introduce the following function: where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$. Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function: +%%RAPH : j'ai coupé la dernière ligne en 2, c'est moche \begin{equation} \begin{array}{lrll} F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} & \longrightarrow & \mathds{B}^{\mathsf{N}} \\ -& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi -(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% +& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+\right.\\ +& & &\left.f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},% \end{array}% \end{equation}% where + and . are the Boolean addition and product operations, and $\overline{x}$ @@ -623,7 +641,7 @@ Consider the phase space: \end{equation} \noindent and the map defined on $\mathcal{X}$: \begin{equation} -G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf} +G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai viré ce label qui existe déjà avant... \end{equation} \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma (S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in @@ -640,7 +658,7 @@ X^{k+1}=G_{f}(X^k).% \right. \end{equation}% -Another time, a shift function appears as a component of these general chaotic +Once more, a shift function appears as a component of these general chaotic iterations. To study the Devaney's chaos property, a distance between two points @@ -650,17 +668,21 @@ Let us introduce: d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}), \label{nouveau d} \end{equation} -\noindent where -\begin{equation} -\left\{ -\begin{array}{lll} -\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}% -}\delta (E_{k},\check{E}_{k})}\textrm{ is another time the Hamming distance}, \\ -\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}% -\sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.% -\end{array}% -\right. -\end{equation} +\noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}% + }\delta (E_{k},\check{E}_{k})}$ is once more the Hamming distance, and +$ \displaystyle{d_{s}(S,\check{S})} = \displaystyle{\dfrac{9}{\mathsf{N}}% + \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$, +%%RAPH : ici, j'ai supprimé tous les sauts à la ligne +%% \begin{equation} +%% \left\{ +%% \begin{array}{lll} +%% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}% +%% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\ +%% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}% +%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.% +%% \end{array}% +%% \right. +%% \end{equation} where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$. @@ -671,7 +693,7 @@ The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$. \begin{proof} $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance -too, thus $d$ will be a distance as sum of two distances. +too, thus $d$, as being the sum of two distances, will also be a distance. \begin{itemize} \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then @@ -688,7 +710,7 @@ inequality is obtained. Before being able to study the topological behavior of the general -chaotic iterations, we must firstly establish that: +chaotic iterations, we must first establish that: \begin{proposition} For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on @@ -724,7 +746,7 @@ so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two poin G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to 0. Let $\varepsilon >0$. \medskip \begin{itemize} -\item If $\varepsilon \geqslant 1$, we see that distance +\item If $\varepsilon \geqslant 1$, we see that the distance between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state). \medskip @@ -741,12 +763,14 @@ G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $% 10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline In conclusion, -$$ +%%RAPH : ici j'ai rajouté une ligne +\begin{flushleft}$$ \forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}% -,\forall n\geqslant N_{0}, - d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) +,\forall n\geqslant N_{0},$$ +$$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right) \leqslant \varepsilon . $$ +\end{flushleft} $G_{f}$ is consequently continuous. \end{proof} @@ -786,7 +810,11 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties claimed in the lemma. \end{proof} -We can now prove the Theorem~\ref{t:chaos des general}... +<<<<<<< HEAD +We can now prove the Theorem~\ref{t:chaos des general}. +======= +We can now prove Theorem~\ref{t:chaos des general}... +>>>>>>> e55d237aba022a66cc2d7650d295b29169878f45 \begin{proof}[Theorem~\ref{t:chaos des general}] Firstly, strong transitivity implies transitivity. @@ -804,8 +832,10 @@ and $t_2\in\mathds{N}$ such that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$. Consider the strategy $\tilde S$ that alternates the first $t_1$ terms -of $S$ and the first $t_2$ terms of $S'$: $$\tilde -S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It +of $S$ and the first $t_2$ terms of $S'$: +%%RAPH : j'ai coupé la ligne en 2 +$$\tilde +S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after $t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic point. Since $\tilde S_t=S_t$ for $t>$32)} in order to obtain the 32 most significant bits of \texttt{t}. -So producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers +Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers that are provided by 3 64-bits PRNGs. This version successfully passes the stringent BigCrush battery of tests~\cite{LEcuyerS07}. @@ -906,7 +939,7 @@ a program similar to the one presented in Listing do so, we must firstly recall that in the CUDA~\cite{Nvid10} environment, threads have a local identifier called \texttt{ThreadIdx}, which is relative to the block containing -them. Furthermore, in CUDA, parts of the code that are executed by the GPU are +them. Furthermore, in CUDA, parts of the code that are executed by the GPU, are called {\it kernels}. @@ -914,10 +947,10 @@ called {\it kernels}. It is possible to deduce from the CPU version a quite similar version adapted to GPU. -The simple principle consists to make each thread of the GPU computing the CPU version of our PRNG. +The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG. Of course, the three xor-like PRNGs used in these computations must have different parameters. -In a given thread, these lasts are +In a given thread, these parameters are randomly picked from another PRNGs. The initialization stage is performed by the CPU. To do it, the ISAAC PRNG~\cite{Jenkins96} is used to set all the @@ -930,8 +963,9 @@ number $x$ that saves the last generated pseudorandom number. Additionally implementation of the xor128, the xorshift, and the xorwow respectively require 4, 5, and 6 unsigned long as internal variables. -\begin{algorithm} +\begin{algorithm} +\begin{small} \KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like PRNGs in global memory\; NumThreads: number of threads\;} @@ -944,14 +978,16 @@ NumThreads: number of threads\;} } store internal variables in InternalVarXorLikeArray[threadIdx]\; } - +\end{small} \caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations} \label{algo:gpu_kernel} \end{algorithm} + + Algorithm~\ref{algo:gpu_kernel} presents a naive implementation of the proposed PRNG on GPU. Due to the available memory in the GPU and the number of threads -used simultenaously, the number of random numbers that a thread can generate +used simultaneously, the number of random numbers that a thread can generate inside a kernel is limited (\emph{i.e.}, the variable \texttt{n} in algorithm~\ref{algo:gpu_kernel}). For instance, if $100,000$ threads are used and if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}, @@ -962,14 +998,14 @@ and the pseudorandom numbers generated by our PRNG, is equal to $100,000\ This generator is able to pass the whole BigCrush battery of tests, for all the versions that have been tested depending on their number of threads -(called \texttt{NumThreads} in our algorithm, tested until $10$ millions). +(called \texttt{NumThreads} in our algorithm, tested up to $5$ million). \begin{remark} -The proposed algorithm has the advantage to manipulate independent +The proposed algorithm has the advantage of manipulating independent PRNGs, so this version is easily adaptable on a cluster of computers too. The only thing to ensure is to use a single ISAAC PRNG. To achieve this requirement, a simple solution consists in using a master node for the initialization. This master node computes the initial parameters -for all the differents nodes involves in the computation. +for all the different nodes involved in the computation. \end{remark} \subsection{Improved Version for GPU} @@ -992,10 +1028,10 @@ been chosen. In practice, we use the xor128 proposed in~\cite{Marsaglia2003} in which unsigned longs (64 bits) have been replaced by unsigned integers (32 bits). -This version also can pass the whole {\it BigCrush} battery of tests. +This version can also pass the whole {\it BigCrush} battery of tests. \begin{algorithm} - +\begin{small} \KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs in global memory\; NumThreads: Number of threads\; @@ -1017,7 +1053,7 @@ array\_comb1, array\_comb2: Arrays containing combinations of size combination\_ } store internal variables in InternalVarXorLikeArray[threadId]\; } - +\end{small} \caption{Main kernel for the chaotic iterations based PRNG GPU efficient version\label{IR}} \label{algo:gpu_kernel2} @@ -1034,7 +1070,7 @@ and two values previously obtained by two other threads). To be certain that we are in the framework of Theorem~\ref{t:chaos des general}, we must guarantee that this dynamical system iterates on the space $\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$. -The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$. +The left term $x$ obviously belongs to $\mathds{B}^ \mathsf{N}$. To prevent from any flaws of chaotic properties, we must check that the right term (the last $t$), corresponding to the strategies, can possibly be equal to any integer of $\llbracket 1, \mathsf{N} \rrbracket$. @@ -1071,7 +1107,7 @@ into the GPU memory has been removed. This step is time consuming and slows down generation. Moreover this storage is completely useless, in case of applications that consume the pseudorandom numbers directly after generation. We can see that when the number of threads is greater -than approximately 30,000 and lower than 5 millions, the number of pseudorandom numbers generated +than approximately 30,000 and lower than 5 million, the number of pseudorandom numbers generated per second is almost constant. With the naive version, this value ranges from 2.5 to 3GSamples/s. With the optimized version, it is approximately equal to 20GSamples/s. Finally we can remark that both GPU cards are quite similar, but in @@ -1082,7 +1118,7 @@ As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of \begin{figure}[htbp] \begin{center} - \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf} + \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf} \end{center} \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG} \label{fig:time_xorlike_gpu} @@ -1096,12 +1132,12 @@ In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized BBS-based PRNG on GPU. On the Tesla C1060 we obtain approximately 700MSample/s and on the GTX 280 about 670MSample/s, which is obviously slower than the xorlike-based PRNG on GPU. However, we will show in the next sections that this -new PRNG has a strong level of security, which is necessary paid by a speed +new PRNG has a strong level of security, which is necessarily paid by a speed reduction. \begin{figure}[htbp] \begin{center} - \includegraphics[scale=.7]{curve_time_bbs_gpu.pdf} + \includegraphics[width=\columnwidth]{curve_time_bbs_gpu.pdf} \end{center} \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG} \label{fig:time_bbs_gpu} @@ -1109,7 +1145,7 @@ reduction. All these experiments allow us to conclude that it is possible to generate a very large quantity of pseudorandom numbers statistically perfect with the xor-like version. -In a certain extend, it is the case too with the secure BBS-based version, the speed deflation being +To a certain extend, it is also the case with the secure BBS-based version, the speed deflation being explained by the fact that the former version has ``only'' chaotic properties and statistical perfection, whereas the latter is also cryptographically secure, as it is shown in the next sections. @@ -1129,17 +1165,17 @@ In this section the concatenation of two strings $u$ and $v$ is classically denoted by $uv$. In a cryptographic context, a pseudorandom generator is a deterministic algorithm $G$ transforming strings into strings and such that, for any -seed $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size -$\ell_G(k)$ with $\ell_G(k)>k$. +seed $s$ of length $m$, $G(s)$ (the output of $G$ on the input $s$) has size +$\ell_G(m)$ with $\ell_G(m)>m$. The notion of {\it secure} PRNGs can now be defined as follows. \begin{definition} A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time algorithm $D$, for any positive polynomial $p$, and for all sufficiently -large $k$'s, -$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$ +large $m$'s, +$$| \mathrm{Pr}[D(G(U_m))=1]-Pr[D(U_{\ell_G(m)})=1]|< \frac{1}{p(m)},$$ where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the -probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the +probabilities are taken over $U_m$, $U_{\ell_G(m)}$ as well as over the internal coin tosses of $D$. \end{definition} @@ -1148,7 +1184,7 @@ distinguish a perfect uniform random generator from $G$ with a non negligible probability. The interested reader is referred to~\cite[chapter~3]{Goldreich} for more information. Note that it is quite easily possible to change the function $\ell$ into any polynomial -function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}. +function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}. The generation schema developed in (\ref{equation Oplus}) is based on a pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume, @@ -1159,7 +1195,7 @@ strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concate the $S_i$'s). The cryptographic PRNG $X$ defined in (\ref{equation Oplus}) is the algorithm mapping any string of length $2N$ $x_0S_0$ into the string $(x_0\oplus S_0 \oplus S_1)(x_0\oplus S_0 \oplus S_1\oplus S_2)\ldots -(x_o\bigoplus_{i=0}^{i=k}S_i)$. Particularly one has $\ell_{X}(2N)=kN=\ell_H(N)$. +(x_o\bigoplus_{i=0}^{i=k}S_i)$. One in particular has $\ell_{X}(2N)=kN=\ell_H(N)$. We claim now that if this PRNG is secure, then the new one is secure too. @@ -1203,8 +1239,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. It follows, by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$ is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}), one has +$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and, +therefore, \begin{equation}\label{PCH-2} -\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1]. +\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1]. \end{equation} Now, using (\ref{PCH-1}) again, one has for every $x$, @@ -1213,7 +1251,7 @@ D^\prime(H(x))=D(\varphi_y(H(x))), \end{equation} where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$, thus -\begin{equation}\label{PCH-3} +\begin{equation}%\label{PCH-3} %%RAPH : j'ai viré ce label qui existe déjà, il est 3 ligne avant D^\prime(H(x))=D(yx), \end{equation} where $y$ is randomly generated. @@ -1223,11 +1261,11 @@ It follows that \mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1]. \end{equation} From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that -there exist a polynomial time probabilistic +there exists a polynomial time probabilistic algorithm $D^\prime$, a positive polynomial $p$, such that for all $k_0$ there exists $N\geq \frac{k_0}{2}$ satisfying $$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$ -proving that $H$ is not secure, a contradiction. +proving that $H$ is not secure, which is a contradiction. \end{proof} @@ -1256,21 +1294,21 @@ lesser than $2^{16}$. So in practice we can choose prime numbers around indistinguishable bits is lesser than or equals to $log_2(log_2(M))$). In other words, to generate a 32-bits number, we need to use 8 times the BBS algorithm with possibly different combinations of $M$. This -approach is not sufficient to be able to pass all the TestU01, +approach is not sufficient to be able to pass all the tests of TestU01, as small values of $M$ for the BBS lead to - small periods. So, in order to add randomness we proceed with + small periods. So, in order to add randomness we have proceeded with the followings modifications. \begin{itemize} \item Firstly, we define 16 arrangement arrays instead of 2 (as described in Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of -the PRNG kernels. In practice, the selection of combinations +the PRNG kernels. In practice, the selection of combination arrays to be used is different for all the threads. It is determined by using the three last bits of two internal variables used by BBS. %This approach adds more randomness. In Algorithm~\ref{algo:bbs_gpu}, character \& is for the bitwise AND. Thus using \&7 with a number -gives the last 3 bits, providing so a number between 0 and 7. +gives the last 3 bits, thus providing a number between 0 and 7. \item Secondly, after the generation of the 8 BBS numbers for each thread, we have a 32-bits number whose period is possibly quite small. So @@ -1278,7 +1316,7 @@ to add randomness, we generate 4 more BBS numbers to shift the 32-bits numbers, and add up to 6 new bits. This improvement is described in Algorithm~\ref{algo:bbs_gpu}. In practice, the last 2 bits of the first new BBS number are used to make a left shift of at most -3 bits. The last 3 bits of the second new BBS number are add to the +3 bits. The last 3 bits of the second new BBS number are added to the strategy whatever the value of the first left shift. The third and the fourth new BBS numbers are used similarly to apply a new left shift and add 3 new bits. @@ -1291,7 +1329,7 @@ variable for BBS number 8 is stored in place 1. \end{itemize} \begin{algorithm} - +\begin{small} \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS in global memory\; NumThreads: Number of threads\; @@ -1327,7 +1365,7 @@ array\_shift[4]=\{0,1,3,7\}\; } store internal variables in InternalVarXorLikeArray[threadId] using a rotation\; } - +\end{small} \caption{main kernel for the BBS based PRNG GPU} \label{algo:bbs_gpu} \end{algorithm} @@ -1345,7 +1383,7 @@ variability. In these operations, we make twice a left shift of $t$ of \emph{at most} 3 bits, represented by \texttt{shift} in the algorithm, and we put \emph{exactly} the \texttt{shift} last bits from a BBS into the \texttt{shift} last bits of $t$. For this, an array named \texttt{array\_shift}, containing the -correspondance between the shift and the number obtained with \texttt{shift} 1 +correspondence between the shift and the number obtained with \texttt{shift} 1 to make the \texttt{and} operation is used. For example, with a left shift of 0, we make an and operation with 0, with a left shift of 3, we make an and operation with 7 (represented by 111 in binary mode). @@ -1402,7 +1440,7 @@ When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ \item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$. \item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$. \item She recomputes the bit-vector $b$ by using BBS and $x_0$. -\item Alice computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. +\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. \end{enumerate} @@ -1415,14 +1453,16 @@ Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too her new public key will be $(S^0, N)$. To encrypt his message, Bob will compute -\begin{equation} -c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right) -\end{equation} +%%RAPH : ici, j'ai mis un simple $ +%\begin{equation} +$c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.$ +$ \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)$ +%%\end{equation} instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. The same decryption stage as in Blum-Goldwasser leads to the sequence $\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$. -Thus, with a simple use of $S^0$, Alice can obtained the plaintext. +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. @@ -1433,7 +1473,7 @@ In this paper, a formerly proposed PRNG based on chaotic iterations has been generalized to improve its speed. It has been proven to be chaotic according to Devaney. Efficient implementations on GPU using xor-like PRNGs as input generators -shown that a very large quantity of pseudorandom numbers can be generated per second (about +have shown that a very large quantity of pseudorandom numbers can be generated per second (about 20Gsamples/s), and that these proposed PRNGs succeed to pass the hardest battery in TestU01, namely the BigCrush. Furthermore, we have shown that when the inputted generator is cryptographically