-%\documentclass{article}
-\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+\documentclass{article}
+%\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+%\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}
% Pour faire des sous-figures dans les figures
\usepackage{subfigure}
-\usepackage{color}
\newtheorem{notation}{Notation}
\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
-\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}}
-
+\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}
+
+\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
+
-\IEEEcompsoctitleabstractindextext{
+%\begin{frontmatter}
+%\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
\end{abstract}
-}
+%}
+%\begin{keyword}
+% pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
+%\end{keyword}
+%\end{frontmatter}
-\maketitle
-\IEEEdisplaynotcompsoctitleabstractindextext
-\IEEEpeerreviewmaketitle
+%\IEEEdisplaynotcompsoctitleabstractindextext
+%\IEEEpeerreviewmaketitle
\section{Introduction}
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 able to withstand malicious
+On the other hand, speed is not the main requirement in cryptography: the most
+important aspect 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. 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.
+
+
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.
-Their desire is to map a given chaotic dynamics into a sequence that seems random
+The main idea is to take advantage from a chaotic dynamical system to obtain a
+generator that is unpredictable, disordered, sensible to its seed, or in other words chaotic.
+Their goal is to map a given chaotic dynamics into a sequence that seems random
and unassailable due to chaos.
-However, the chaotic maps used as a pattern are defined in the real line
+However, the chaotic maps used as patterns are defined in the real line
whereas computers deal with finite precision numbers.
This distortion leads to a deflation of both chaotic properties and speed.
Furthermore, authors of such chaotic generators often claim their PRNG
-as secure due to their chaos properties, but there is no obvious relation
+are secure due to their chaos properties, but there is no obvious relation
between chaos and security as it is understood in cryptography.
This is why the use of chaos for PRNG still remains marginal and disputable.
of a PRNG. But they are not substitutable for security or statistical perfection.
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
+to a statistically deflective PRNG, in order to improve its statistical
properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.
On the other hand, chaos can be added to a fast, statistically perfect PRNG and/or a
cryptographically secure one, in case where chaos can be of interest,
{\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}.
+More precisely, each time we performed a test on a PRNG, we ran it
+twice in order to observe if all $p-$values were 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
+[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
proposed generator, without any lack of chaos or statistical properties.
In particular, a version of this PRNG on graphics processing units (GPU)
is proposed.
-Although GPU was initially designed to accelerate
+Although GPUs were initially designed to accelerate
the manipulation of images, they are nowadays commonly used in many scientific
applications. Therefore, it is important to be able to generate pseudorandom
-numbers inside a GPU when a scientific application runs in it. This remark
+numbers inside a GPU when it is run by 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 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
+cryptographical security of the inputted PRNG, when the latter has such a
property.
Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
key encryption protocol by using the proposed method.
+
+{\bf Main contributions.} In this paper a new PRNG using chaotic iteration
+is defined. From a theoretical point of view, it is proven that it has fine
+topological chaotic properties and that it is cryptographically secure (when
+the initial PRNG is also cryptographically secure). From a practical point of
+view, experiments point out a very good statistical behavior. An optimized
+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
+\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.
+
+
+
+
The remainder of this paper is organized as follows. In Section~\ref{section:related
works} we review some GPU implementations of PRNGs. Section~\ref{section:BASIC
RECALLS} gives some basic recalls on the well-known Devaney's formulation of chaos,
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
- 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
-generator is cryptographically secure, then it is the case too for the
+generator is cryptographically secure, then it is also the case of the
generator provided by the post-treatment.
+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
+in Section~\ref{sec:CSGPU} 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.
-\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
literature, so that exhaustivity is impossible.
-This is why authors of this document only give reference to the most significant attempts
+This is why the authors of this document only only refer 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
only when the information is given in the related work.
1MSample/s whereas a billion numbers per second is 1GSample/s.
In \cite{Pang:2008:cec} a PRNG based on cellular automata is defined
-with no requirement to an high precision integer arithmetic or to any bitwise
+with no requirement to a high precision integer arithmetic or to any bitwise
operations. Authors can generate about
3.2MSamples/s on a GeForce 7800 GTX GPU, which is quite an old card now.
However, there is neither a mention of statistical tests nor any proof of
Authors of~\cite{conf/fpga/ThomasHL09} have studied the implementation of some
PRNGs on different computing architectures: CPU, field-programmable gate array
-(FPGA), massively parallel processors, and GPU. This study is of interest, because
+(FPGA), massively parallel processors, and GPU. This study is interesting, because
the performance of the same PRNGs on different architectures are compared.
FPGA appears as the fastest and the most
efficient architecture, providing the fastest number of generated pseudorandom numbers
per joule.
-However, we notice that authors can ``only'' generate between 11 and 16GSamples/s
+However, we notice that the authors can ``only'' generate between 11 and 16GSamples/s
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
other things
Xorwow~\cite{Marsaglia2003} and some variants of Sobol. The tests reported show that
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).
+But their PRNGs cannot pass the whole TestU01 battery (only one test has failed).
\newline
\newline
-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.
+We can finally remark that, to the best of our knowledge, no GPU implementation has ever been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered.
\section{Basic Recalls}
\label{section:BASIC RECALLS}
\subsection{Devaney's Chaotic Dynamical Systems}
-
+\label{subsec:Devaney}
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
is for the $k^{th}$ composition of a function $f$. Finally, the following
\mathds{N}^*$ of elements (or \emph{cells}), so that each cell has a
Boolean \emph{state}. Having $\mathsf{N}$ Boolean values for these
cells leads to the definition of a particular \emph{state of the
-system}. A sequence which elements belong to $\llbracket 1;\mathsf{N}
+system}. A sequence whose elements belong to $\llbracket 1;\mathsf{N}
\rrbracket $ is called a \emph{strategy}. The set of all strategies is
denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$
We have proposed in~\cite{bgw09:ip} a new family of generators that receives
two PRNGs as inputs. These two generators are mixed with chaotic iterations,
leading thus to a new PRNG that
-\begin{color}{red}
-should improves the statistical properties of each
+should improve the statistical properties of each
generator taken alone.
-Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input
-present.
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as input present.
In order to make the Old CI PRNG usable in practice, we have proposed
an adapted version of the chaotic iteration based generator in~\cite{bg10:ip}.
-In this ``New CI PRNG'', we prevent from changing twice a given
-bit between two outputs.
+In this ``New CI PRNG'', we prevent a given bit from changing twice between two outputs.
This new generator is designed by the following process.
First of all, some chaotic iterations have to be done to generate a sequence
$\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$
of Boolean vectors, which are the successive states of the iterated system.
-Some of these vectors will be randomly extracted and our pseudo-random bit
+Some of these vectors will be randomly extracted and our pseudorandom bit
flow will be constituted by their components. Such chaotic iterations are
realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean
vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in
Then, at each iteration, only the $S^n$-th component of state $x^n$ is
updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$.
-Such a procedure is equivalent to achieve chaotic iterations with
+Such a procedure is equivalent to achieving chaotic iterations with
the Boolean vectorial negation $f_0$ and some well-chosen strategies.
Finally, some $x^n$ are selected
-by a sequence $m^n$ as the pseudo-random bit sequence of our generator.
+by a sequence $m^n$ as the pseudorandom bit sequence of our generator.
$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers.
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 are 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}
}
\ENDFOR
\STATE$a\leftarrow{PRNG_1()}$\;
-\STATE$m\leftarrow{g(a)}$\;
-\STATE$k\leftarrow{m}$\;
+\STATE$k\leftarrow{g(a)}$\;
\WHILE{$i=0,\dots,k$}
\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
\label{Chaotic iteration1}
\end{algorithmic}
\end{algorithm}
-\end{color}
+
+
+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 impressive,
+as the chaotic iterations post-treatment fails with only 8 tests of the TestU01 battery.
+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,\begin{color}{red} we now propose to choose a
-subset of components and to update them together, for speed improvements. Such a proposition leads\end{color}
+Instead of updating only one cell at each iteration, we now propose to choose a
+subset of components and to update them together, for speed improvement. Such a proposition leads
to a kind of merger of the two sequences used in Algorithms
\ref{CI Algorithm} and \ref{Chaotic iteration1}. When the updating function is the vectorial negation,
this algorithm can be rewritten as follows:
\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
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 on the fact that, after $p$ iterations,
+the state of the system may or not be the same as before these iterations.
+\end{color}
-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:
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).
\begin{proof}
$d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
-too, thus $d$, as being the sum of two distances, will also be a distance.
+too, thus $d$, 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
In conclusion,
%%RAPH : ici j'ai rajouté une ligne
+%%TOF : ici j'ai rajouté un commentaire
+%%TOF : ici aussi
$
\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
,$ $\forall n\geqslant N_{0},$
claimed in the lemma.
\end{proof}
-We can now prove the Theorem~\ref{t:chaos des general}.
+We can now prove Theorem~\ref{t:chaos des general}.
\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.
\end{proof}
-\begin{color}{red}
-\section{Statistical Improvements Using Chaotic Iterations}
-
-\label{The generation of pseudo-random sequence}
-
-
-Let us now explain why we are reasonable grounds to believe that chaos
-can improve statistical properties.
-We will show in this section that, when mixing defective PRNGs with
-chaotic iterations, the result presents better statistical properties
-(this section summarizes the work of~\cite{bfg12a:ip}).
-
-\subsection{Details of some Existing Generators}
-
-The list of defective PRNGs we will use
-as inputs for the statistical tests to come is introduced here.
-
-Firstly, the simple linear congruency generator (LCGs) will be used.
-It is defined by the following recurrence:
-\begin{equation}
-x^n = (ax^{n-1} + c)~mod~m
-\label{LCG}
-\end{equation}
-where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than
-$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three)
-combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
+%\section{Statistical Improvements Using Chaotic Iterations}
+
+%\label{The generation of pseudorandom sequence}
+
+
+%Let us now explain why we have reasonable ground to believe that chaos
+%can improve statistical properties.
+%We will show in this section that chaotic properties as defined in the
+%mathematical theory of chaos are related to some statistical tests that can be found
+%in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
+%chaotic iterations, the new generator presents better statistical properties
+%(this section summarizes and extends the work of~\cite{bfg12a:ip}).
+
+
+
+%\subsection{Qualitative relations between topological properties and statistical tests}
+
+
+%There are various relations between topological properties that describe an unpredictable behavior for a discrete
+%dynamical system on the one
+%hand, and statistical tests to check the randomness of a numerical sequence
+%on the other hand. These two mathematical disciplines follow a similar
+%objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
+%recurrent sequence), with two different but complementary approaches.
+%It is true that the following illustrative links give only qualitative arguments,
+%and proofs should be provided later to make such arguments irrefutable. However
+%they give a first understanding of the reason why we think that chaotic properties should tend
+%to improve the statistical quality of PRNGs.
+%%
+%Let us now list some of these relations between topological properties defined in the mathematical
+%theory of chaos and tests embedded into the NIST battery. %Such relations need to be further
+%%investigated, but they presently give a first illustration of a trend to search similar properties in the
+%%two following fields: mathematical chaos and statistics.
+
+
+%\begin{itemize}
+% \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must
+%have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
+%a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
+%is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
+%knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
+%the two following NIST tests~\cite{Nist10}:
+% \begin{itemize}
+% \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
+% \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
+% \end{itemize}
+
+%\item \textbf{Transitivity}. This topological property previously introduced states that the dynamical system is intrinsically complicated: it cannot be simplified into
+%two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space.
+%This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
+%of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
+%is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
+% \begin{itemize}
+% \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
+% \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
+% \end{itemize}
+
+%\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
+%to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
+% \begin{itemize}
+% \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
+% \end{itemize}
+% \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
+%has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
+%rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach,
+%whereas topological entropy is defined as follows:
+%$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
+%leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations,
+%the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f) = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
+%This value measures the average exponential growth of the number of distinguishable orbit segments.
+%In this sense, it measures the complexity of the topological dynamical system, whereas
+%the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
+% \begin{itemize}
+%\item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
+% \end{itemize}
+
+% \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
+%not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
+% \begin{itemize}
+%\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
+%\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
+% \end{itemize}
+%\end{itemize}
+
+
+%We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
+%things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
+%and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
+%where $\mathsf{N}$ is the size of the iterated vector.
+%These topological properties make that we are ground to believe that a generator based on chaotic
+%iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
+%the NIST one. The following subsections, in which we prove that defective generators have their
+%statistical properties improved by chaotic iterations, show that such an assumption is true.
+
+%\subsection{Details of some Existing Generators}
+
+%The list of defective PRNGs we will use
+%as inputs for the statistical tests to come is introduced here.
+
+%Firstly, the simple linear congruency generators (LCGs) will be used.
+%They are defined by the following recurrence:
+%\begin{equation}
+%x^n = (ax^{n-1} + c)~mod~m,
+%\label{LCG}
+%\end{equation}
+%where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to
+%$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three)
+%combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
-Secondly, the multiple recursive generators (MRGs) will be used too, which
-are based on a linear recurrence of order
-$k$, modulo $m$~\cite{LEcuyerS07}:
-\begin{equation}
-x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m
-\label{MRG}
-\end{equation}
-Combination of two MRGs (referred as 2MRGs) is also used in these experimentations.
+%Secondly, the multiple recursive generators (MRGs) which will be used,
+%are based on a linear recurrence of order
+%$k$, modulo $m$~\cite{LEcuyerS07}:
+%\begin{equation}
+%x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .
+%\label{MRG}
+%\end{equation}
+%The combination of two MRGs (referred as 2MRGs) is also used in these experiments.
-Generators based on linear recurrences with carry will be regarded too.
-This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
-\begin{equation}
-\label{AWC}
-\begin{array}{l}
-x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
-c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
-the SWB generator, having the recurrence:
-\begin{equation}
-\label{SWB}
-\begin{array}{l}
-x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
-c^n=\left\{
-\begin{array}{l}
-1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
-0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
-and the SWC generator designed by R. Couture, which is based on the following recurrence:
-\begin{equation}
-\label{SWC}
-\begin{array}{l}
-x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
-c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
+%Generators based on linear recurrences with carry will be regarded too.
+%This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
+%\begin{equation}
+%\label{AWC}
+%\begin{array}{l}
+%x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
+%c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
+%the SWB generator, having the recurrence:
+%\begin{equation}
+%\label{SWB}
+%\begin{array}{l}
+%x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
+%c^n=\left\{
+%\begin{array}{l}
+%1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
+%0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
+%and the SWC generator, which is based on the following recurrence:
+%\begin{equation}
+%\label{SWC}
+%\begin{array}{l}
+%x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
+%c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
-Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
-\begin{equation}
-x^n = x^{n-r} \oplus x^{n-k}
-\label{GFSR}
-\end{equation}
+%Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
+%\begin{equation}
+%x^n = x^{n-r} \oplus x^{n-k} .
+%\label{GFSR}
+%\end{equation}
-Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is:
+%Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
-\begin{equation}
-\label{INV}
-\begin{array}{l}
-x^n=\left\{
-\begin{array}{ll}
-(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
-a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
-
-
-
-\begin{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test}
-\label{TestU011}
-\centering
- \begin{tabular}{lccccc}
- \toprule
-Test name &Tests& Logistic & XORshift & ISAAC\\
-Rabbit & 38 &21 &14 &0 \\
-Alphabit & 17 &16 &9 &0 \\
-Pseudo DieHARD &126 &0 &2 &0 \\
-FIPS\_140\_2 &16 &0 &0 &0 \\
-SmallCrush &15 &4 &5 &0 \\
-Crush &144 &95 &57 &0 \\
-Big Crush &160 &125 &55 &0 \\ \hline
-Failures & &261 &146 &0 \\
-\bottomrule
- \end{tabular}
-\end{table}
-
-
-
-\begin{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)}
-\label{TestU01 for Old CI}
-\centering
- \begin{tabular}{lcccc}
- \toprule
-\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
-&Logistic& XORshift& ISAAC&ISAAC \\
-&+& +& + & + \\
-&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5}
-Rabbit &7 &2 &0 &0 \\
-Alphabit & 3 &0 &0 &0 \\
-DieHARD &0 &0 &0 &0 \\
-FIPS\_140\_2 &0 &0 &0 &0 \\
-SmallCrush &2 &0 &0 &0 \\
-Crush &47 &4 &0 &0 \\
-Big Crush &79 &3 &0 &0 \\ \hline
-Failures &138 &9 &0 &0 \\
-\bottomrule
- \end{tabular}
-\end{table}
-
-
-
-
-
-\subsection{Statistical tests}
-\label{Security analysis}
-
-Three batteries of tests are reputed and usually used
-to evaluate the statistical properties of newly designed pseudorandom
-number generators. These batteries are named DieHard~\cite{Marsaglia1996},
-the NIST suite~\cite{ANDREW2008}, and the most stringent one called
-TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
-
-
-
-\label{Results and discussion}
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
-\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
- \hline\hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
-NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
-DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
-
-Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the
-results on the two firsts batteries recalled above, indicating that all the PRNGs presented
-in the previous section
-cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot
-fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
-iterations can solve this issue.
-%More precisely, to
-%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
-%\begin{enumerate}
-% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
-% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
-% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
%\begin{equation}
+%\label{INV}
%\begin{array}{l}
-%\left\{
-%\begin{array}{l}
-%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
-%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
-%\end{equation}
-%$m$ is called the \emph{functional power}.
-%\end{enumerate}
-%
-The obtained results are reproduced in Table
-\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
-The scores written in boldface indicate that all the tests have been passed successfully, whereas an
-asterisk ``*'' means that the considered passing rate has been improved.
-The improvements are obvious for both the ``Old CI'' and ``New CI'' generators.
-Concerning the ``Xor CI PRNG'', the speed improvement makes that statistical
-results are not as good as for the two other versions of these CIPRNGs.
-
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
-\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
- \hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
-Old CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
-New CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
-Xor CIPRNG\\ \hline\hline
-NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
-DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
-
-
-We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
-the statistical behavior of the Xor CI version by combining more than one
-$\oplus$ operation. Results are summarized in~\ref{threshold}, showing
-that rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
-using chaotic iterations on defective generators.
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
-\label{threshold}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
- \hline
-Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
-Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
-\end{tabular}
-\end{table*}
-
-Finally, the TestU01 battery as been launched on three well-known generators
-(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
-see Table~\ref{TestU011}). These results can be compared with
-Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
-Old CI PRNG that has received these generators.
-
-
-Next subsection gives a concrete implementation of this Xor CI PRNG, which will
-new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does not
-raise ambiguity.
-\end{color}
-
-\subsection{Efficient Implementation of a PRNG based on Chaotic Iterations}
+%x^n=\left\{
+%\begin{array}{ll}
+%(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
+%a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
+
+
+
+%\begin{table}
+%%\renewcommand{\arraystretch}{1}
+%\caption{TestU01 Statistical Test Failures}
+%\label{TestU011}
+%\centering
+% \begin{tabular}{lccccc}
+% \toprule
+%Test name &Tests& Logistic & XORshift & ISAAC\\
+%Rabbit & 38 &21 &14 &0 \\
+%Alphabit & 17 &16 &9 &0 \\
+%Pseudo DieHARD &126 &0 &2 &0 \\
+%FIPS\_140\_2 &16 &0 &0 &0 \\
+%SmallCrush &15 &4 &5 &0 \\
+%Crush &144 &95 &57 &0 \\
+%Big Crush &160 &125 &55 &0 \\ \hline
+%Failures & &261 &146 &0 \\
+%\bottomrule
+% \end{tabular}
+%\end{table}
+
+
+
+%\begin{table}
+%%\renewcommand{\arraystretch}{1}
+%\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
+%\label{TestU01 for Old CI}
+%\centering
+% \begin{tabular}{lcccc}
+% \toprule
+%\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
+%&Logistic& XORshift& ISAAC&ISAAC \\
+%&+& +& + & + \\
+%&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5}
+%Rabbit &7 &2 &0 &0 \\
+%Alphabit & 3 &0 &0 &0 \\
+%DieHARD &0 &0 &0 &0 \\
+%FIPS\_140\_2 &0 &0 &0 &0 \\
+%SmallCrush &2 &0 &0 &0 \\
+%Crush &47 &4 &0 &0 \\
+%Big Crush &79 &3 &0 &0 \\ \hline
+%Failures &138 &9 &0 &0 \\
+%\bottomrule
+% \end{tabular}
+%\end{table}
+
+
+
+
+
+%\subsection{Statistical tests}
+%\label{Security analysis}
+
+%Three batteries of tests are reputed and regularly used
+%to evaluate the statistical properties of newly designed pseudorandom
+%number generators. These batteries are named DieHard~\cite{Marsaglia1996},
+%the NIST suite~\cite{ANDREW2008}, and the most stringent one called
+%TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
+
+
+
+%\label{Results and discussion}
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1}
+%\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
+%\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
+% \hline\hline
+%Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
+%NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
+%DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
+%\end{tabular}
+%\end{table*}
+
+%Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the
+%results on the two first batteries recalled above, indicating that all the PRNGs presented
+%in the previous section
+%cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot
+%fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
+%iterations can solve this issue.
+%%More precisely, to
+%%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
+%%\begin{enumerate}
+%% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
+%% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
+%% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%%\begin{equation}
+%%\begin{array}{l}
+%%\left\{
+%%\begin{array}{l}
+%%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
+%%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
+%%\end{equation}
+%%$m$ is called the \emph{functional power}.
+%%\end{enumerate}
+%%
+%The obtained results are reproduced in Table
+%\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
+%The scores written in boldface indicate that all the tests have been passed successfully, whereas an
+%asterisk ``*'' means that the considered passing rate has been improved.
+%The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
+%Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
+% are not as good as for the two other versions of these CIPRNGs.
+%However 8 tests have been improved (with no deflation for the other results).
+
+
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1.3}
+%\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
+%\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
+% \hline
+%Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
+%Old CIPRNG\\ \hline \hline
+%NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
+%New CIPRNG\\ \hline \hline
+%NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
+%Xor CIPRNG\\ \hline\hline
+%NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
+%DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
+%\end{tabular}
+%\end{table*}
+
+
+%We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
+%the statistical behavior of the Xor CI version by combining more than one
+%$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
+%the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
+%Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
+%using chaotic iterations on defective generators.
+
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1.3}
+%\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
+%\label{threshold}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
+% \hline
+%Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
+%Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
+%\end{tabular}
+%\end{table*}
+
+%Finally, the TestU01 battery has been launched on three well-known generators
+%(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
+%see Table~\ref{TestU011}). These results can be compared with
+%Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
+%Old CI PRNG that has received these generators.
+%The obvious improvement speaks for itself, and together with the other
+%results recalled in this section, it reinforces the opinion that a strong
+%correlation between topological properties and statistical behavior exists.
+
+
+%The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
+%fastest generator in the chaotic iteration based family. In the remainder,
+%this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
+%raise ambiguity.
+
+
+\section{Toward Efficiency and Improvement for CI PRNG}
\label{sec:efficient PRNG}
+
+\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
%
%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}.
-\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}
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}.
+stringent BigCrush battery of tests~\cite{LEcuyerS07}.
+At this point, we thus
+have defined an efficient and statistically unbiased generator. Its speed is
+directly related to the use of linear operations, but for the same reason,
+this fast generator cannot be proven as secure.
-\section{Efficient PRNGs based on Chaotic Iterations on GPU}
+
+
+\subsection{Efficient PRNGs based on Chaotic Iterations on GPU}
\label{sec:efficient PRNG gpu}
In order to take benefits from the computing power of GPU, a program
needs to have independent blocks of threads that can be computed
simultaneously. In general, the larger the number of threads is, the
more local memory is used, and the less branching instructions are
-used (if, while, ...), the better the performances on GPU is.
+used (if, while, ...), the better the performances on GPU are.
Obviously, having these requirements in mind, it is possible to build
a program similar to the one presented in Listing
\ref{algo:seqCIPRNG}, which computes pseudorandom numbers on GPU. To
\subsection{Naive Version for GPU}
-It is possible to deduce from the CPU version a quite similar version adapted to GPU.
+It is possible to deduce from the CPU version a fairly similar version adapted to GPU.
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 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
+To do so, the ISAAC PRNG~\cite{Jenkins96} is used to set all the
parameters embedded into each thread.
The implementation of the three
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
\subsection{Improved Version for GPU}
-As GPU cards using CUDA have shared memory between threads of the same block, it
+As GPU cards using CUDA have a shared memory between threads of the same block, it
is possible to use this feature in order to simplify the previous algorithm,
i.e., to use less than 3 xor-like PRNGs. The solution consists in computing only
one xor-like PRNG by thread, saving it into the shared memory, and then to use the results
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
\label{algo:gpu_kernel2}
\end{algorithm}
-\subsection{Theoretical Evaluation of the Improved Version}
+\subsection{Chaos Evaluation of the Improved Version}
A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having
the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
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 to $\mathds{B}^ \mathsf{N}$.
-To prevent from any flaws of chaotic properties, we must check that the right
+To prevent 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 $\llbracket 1, \mathsf{N} \rrbracket$.
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
practice, the Tesla C1060 has more memory than the GTX 280, and this memory
-should be of better quality.
+is of better quality.
As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of about
138MSample/s when using one core of the Xeon E5530.
\begin{figure}[htbp]
\begin{center}
- \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf}
+ \includegraphics[scale=0.7]{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}
\begin{figure}[htbp]
\begin{center}
- \includegraphics[width=\columnwidth]{curve_time_bbs_gpu.pdf}
+ \includegraphics[scale=0.7]{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}
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.
+as shown in the next sections.
\section{Security Analysis}
-\label{sec:security analysis}
+This section is dedicated to the security analysis of the
+ proposed PRNGs, both from a theoretical and from a practical point of view.
+
+\subsection{Theoretical Proof of Security}
+\label{sec:security analysis}
+
+The standard definition
+ of {\it indistinguishability} used 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
+ is a relative notion between breaking time and the sizes of the
+ keys/seeds. Of course, if small keys or seeds are chosen, the system can
+ 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.
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
internal coin tosses of $D$.
\end{definition}
-Intuitively, it means that there is no polynomial time algorithm that can
-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 $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.
+Intuitively, it means that there is no polynomial time algorithm that can
+distinguish a perfect uniform random generator from $G$ with a non negligible
+probability. An equivalent formulation of this well-known security property
+means that it is possible \emph{in practice} to predict the next bit of the
+generator, knowing all the previously produced ones. 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 $\ell^\prime(m)>m)$~\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,
\end{proposition}
\begin{proof}
-The proposition is proved by contraposition. Assume that $X$ is not
+The proposition is proven by contraposition. Assume that $X$ is not
secure. By Definition, there exists a polynomial time probabilistic
algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
$N\geq \frac{k_0}{2}$ satisfying
\end{proof}
+
+\subsection{Practical Security Evaluation}
+\label{sec:Practicak evaluation}
+
+Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when
+they are XORed with an already cryptographically
+secure PRNG. But, as stated previously,
+such a property does not mean that, whatever the
+key size, no attacker can predict the next bit
+knowing all the previously released ones.
+However, given a key size, it is possible to
+measure in practice the minimum duration needed
+for an attacker to break a cryptographically
+secure PRNG, if we know the power of his/her
+machines. Such a concrete security evaluation
+is related to the $(T,\varepsilon)-$security
+notion, which is recalled and evaluated in what
+follows, for the sake of completeness.
+
+Let us firstly recall that,
+\begin{definition}
+Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
+in time $T$.
+Let $\varepsilon > 0$.
+$\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
+generator $G$ if
+
+$$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right. - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$$
+\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.
+\end{definition}
+
+Let us recall that the running time of a probabilistic algorithm is defined to be the
+maximum of the expected number of steps needed to produce an output, maximized
+over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
+We are now able to define the notion of cryptographically secure PRNGs:
+
+\begin{definition}
+A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
+\end{definition}
+
+
+
+
+
+
+
+Suppose now that the PRNG of Eq.~\eqref{equation Oplus} will work during
+$M=100$ time units, and that during this period,
+an attacker can realize $10^{12}$ clock cycles.
+We thus wonder whether, during the PRNG's
+lifetime, the attacker can distinguish this
+sequence from a truly random one, with a probability
+greater than $\varepsilon = 0.2$.
+We consider that $N$ has 900 bits.
+
+Predicting the next generated bit knowing all the
+previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
+next bit in the BBS generator, which
+is cryptographically secure. More precisely, it
+is $(T,\varepsilon)-$secure: no
+$(T,\varepsilon)-$distinguishing attack can be
+successfully realized on this PRNG, if~\cite{Fischlin}
+\begin{equation}
+T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
+\label{mesureConcrete}
+\end{equation}
+where $M$ is the length of the output ($M=100$ in
+our example), and $L(N)$ is equal to
+$$
+2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~ 2))^\frac{2}{3}\right)
+$$
+is the number of clock cycles to factor a $N-$bit
+integer.
+
+
+
+
+A direct numerical application shows that this attacker
+cannot achieve his/her $(10^{12},0.2)$ distinguishing
+attack in that context.
+
+
+
\section{Cryptographical Applications}
\subsection{A Cryptographically Secure PRNG for GPU}
The modulus operation is the most time consuming operation for current
GPU cards. So in order to obtain quite reasonable performances, it is
required to use only modulus on 32-bits integer numbers. Consequently
-$x_n^2$ need to be lesser than $2^{32}$, and thus the number $M$ must be
-lesser than $2^{16}$. So in practice we can choose prime numbers around
+$x_n^2$ need to be inferior than $2^{32}$, and thus the number $M$ must be
+inferior than $2^{16}$. So in practice we can choose prime numbers around
256 that are congruent to 3 modulus 4. With 32-bits numbers, only the
4 least significant bits of $x_n$ can be chosen (the maximum number of
indistinguishable bits is lesser than or equals to
}
\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}
where $S^n$ is referred in this algorithm as $t$: each iteration of this
PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted
by secure bits produced by the BBS generator, and thus, due to
-Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically
-secure.
-
-
-
-\begin{color}{red}
-\subsection{Practical Security Evaluation}
-
-Suppose now that the PRNG will work during
-$M=100$ time units, and that during this period,
-an attacker can realize $10^{12}$ clock cycles.
-We thus wonder whether, during the PRNG's
-lifetime, the attacker can distinguish this
-sequence from truly random one, with a probability
-greater than $\varepsilon = 0.2$.
-We consider that $N$ has 900 bits.
-
-The random process is the BBS generator, which
-is cryptographically secure. More precisely, it
-is $(T,\varepsilon)-$secure: no
-$(T,\varepsilon)-$distinguishing attack can be
-successfully realized on this PRNG, if~\cite{Fischlin}
-$$
-T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
-$$
-where $M$ is the length of the output ($M=100$ in
-our example), and $L(N)$ is equal to
-$$
-2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)
-$$
-is the number of clock cycles to factor a $N-$bit
-integer.
+Proposition~\ref{cryptopreuve}, the resulted PRNG is
+cryptographically secure.
+
+As stated before, even if the proposed PRNG is cryptocaphically
+secure, it does not mean that such a generator
+can be used as described here when attacks are
+awaited. The problem is to determine the minimum
+time required for an attacker, with a given
+computational power, to predict under a probability
+lower than 0.5 the $n+1$th bit, knowing the $n$
+previous ones. The proposed GPU generator will be
+useful in a security context, at least in some
+situations where a secret protected by a pseudorandom
+keystream is rapidly obsolete, if this time to
+predict the next bit is large enough when compared
+to both the generation and transmission times.
+It is true that the prime numbers used in the last
+section are very small compared to up-to-date
+security recommendations. However the attacker has not
+access to each BBS, but to the output produced
+by Algorithm~\ref{algo:bbs_gpu}, which is far
+more complicated than a simple BBS. Indeed, to
+determine if this cryptographically secure PRNG
+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}
+must be established. The authors
+hope to achieve this difficult task in a future
+work.
-A direct numerical application shows that this attacker
-cannot achieve its $(10^{12},0.2)$ distinguishing
-attack in that context.
-
-\end{color}
\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
\label{Blum-Goldwasser}
To encrypt his message, Bob will compute
%%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 $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$.
+\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
-$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.
+$$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right).$$
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.
namely the BigCrush.
Furthermore, we have shown that when the inputted generator is cryptographically
secure, then it is the case too for the PRNG we propose, thus leading to
-the possibility to develop fast and secure PRNGs using the GPU architecture.
-\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it
-behaves chaotically, has finally been proposed. \end{color}
+the possibility of developping fast and secure PRNGs using the GPU architecture.
+An improvement of the Blum-Goldwasser cryptosystem, making it
+behave chaotically, has finally been proposed.
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.
