-%\documentclass{article}
+\documentclass{article}
%\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
-\documentclass[preprint,12pt]{elsarticle}
+%\documentclass[preprint,12pt]{elsarticle}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{fullpage}
\usepackage{tabularx}
\usepackage{multirow}
+\usepackage{color}
+
% Pour mathds : les ensembles IR, IN, etc.
\usepackage{dsfont}
\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
-
+\begin{document}
\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*\\ FEMTO-ST Institute, UMR 6174 CNRS,\\ University of Franche-Comt\'{e}, Besan\c con, France\\ * Authors in alphabetic order}
-
+%% \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}
+
+\author{Christophe Guyeux \and Rapha\"{e}l Couturier \and Pierre-Cyrille Héam \and Jacques M. Bahi\\
+FEMTO-ST Institute, UMR 6174 CNRS,\\ University of Franche Comte, Belfort, France}
+
+\maketitle
+
+
+%\begin{frontmatter}
%\IEEEcompsoctitleabstractindextext{
\begin{abstract}
In this paper we present a new pseudorandom number generator (PRNG) on
\end{abstract}
%}
+%\begin{keyword}
+% pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
+%\end{keyword}
+%\end{frontmatter}
-\maketitle
%\IEEEdisplaynotcompsoctitleabstractindextext
%\IEEEpeerreviewmaketitle
original implementation of this PRNG is also proposed and experimented.
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
+statistical behavior). Experiments are also provided using
+\begin{color}{red} the well-known Blum-Blum-Shub
+(BBS)
+\end{color}
+as the initial
random generator. The generation speed is significantly weaker.
-Note also that an original qualitative comparison between topological chaotic
-properties and statistical tests is also proposed.
+%Note also that an original qualitative comparison between topological chaotic
+%properties and statistical tests is also proposed.
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.
+\begin{color}{red}
+Obviously, when $S$ is periodic of period $p$, then $x$ is periodic too of
+period either $p$ or $2p$, depending of the fact that, after $p$ iterations,
+the state of the system may or not be the same than before these iterations.
+\end{color}
The single basic component presented in Eq.~\ref{equation Oplus} is of
ordinary use as a good elementary brick in various PRNGs. It corresponds
PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
and the pseudorandom numbers generated by our PRNG, is equal to $100,000\times ((4+5+6)\times
2+(1+100))=1,310,000$ 32-bits numbers, that is, approximately $52$Mb.
+\begin{color}{red}
+Remark that the only requirement regarding the seed regarding the security of our PRNG is
+that it must be randomly picked. Indeed, the asymptotic security of BBS guarantees
+that, as the seed length increases, no polynomial time statistical test can
+distinguish the pseudorandom sequences from truly random sequences with non-negligible probability,
+see, \emph{e.g.},~\cite{Sidorenko:2005:CSB:2179218.2179250}.
+\end{color}
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
array\_comb1, array\_comb2: Arrays containing combinations of size combination\_size\;}
\KwOut{NewNb: array containing random numbers in global memory}
-\If{threadId is concerned} {
- retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory and x\;
+\If{threadIdx is concerned} {
+ retrieve data from InternalVarXorLikeArray[threadIdx] in local variables including shared memory and x\;
offset = threadIdx\%combination\_size\;
o1 = threadIdx-offset+array\_comb1[offset]\;
o2 = threadIdx-offset+array\_comb2[offset]\;
\For{i=1 to n} {
t=xor-like()\;
t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;
- shared\_mem[threadId]=t\;
+ shared\_mem[threadIdx]=t\;
x = x\textasciicircum t\;
- store the new PRNG in NewNb[NumThreads*threadId+i]\;
+ store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
}
- store internal variables in InternalVarXorLikeArray[threadId]\;
+ store internal variables in InternalVarXorLikeArray[threadIdx]\;
}
\end{small}
\caption{Main kernel for the chaotic iterations based PRNG GPU efficient
$\mathcal{D}$ 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.
}
\KwOut{NewNb: array containing random numbers in global memory}
-\If{threadId is concerned} {
- retrieve data from InternalVarBBSArray[threadId] in local variables including shared memory and x\;
+\If{threadIdx is concerned} {
+ retrieve data from InternalVarBBSArray[threadIdx] in local variables including shared memory and x\;
we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\;
offset = threadIdx\%combination\_size\;
o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
t$<<$=shift\;
t|=BBS2(bbs2)\&array\_shift[shift]\;
t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;
- shared\_mem[threadId]=t\;
+ shared\_mem[threadIdx]=t\;
x = x\textasciicircum t\;
- store the new PRNG in NewNb[NumThreads*threadId+i]\;
+ store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
}
- store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
+ store internal variables in InternalVarXorLikeArray[threadIdx] using a rotation\;
}
\end{small}
\caption{main kernel for the BBS based PRNG GPU}
In future work we plan to extend this research, building a parallel PRNG for clusters or
grid computing. Topological properties of the various proposed generators will be investigated,
and the use of other categories of PRNGs as input will be studied too. The improvement
-of Blum-Goldwasser will be deepened. Finally, we
+of Blum-Goldwasser will be deepened.
+\begin{color}{red}
+Another aspect to consider might be different accelerator-based systems like
+Intel Xeon Phi cards and speed measurements using such cards: as heterogeneity of
+supercomputers tends to increase using other accelerators than GPGPUs,
+a Xeon Phi solution might be interesting to investigate.
+\end{color}
+ Finally, we
will try to enlarge the quantity of pseudorandom numbers generated per second either
in a simulation context or in a cryptographic one.
