correct papier

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 5ebe0efb295e6ae78a21bead73fd1cf8cef19b52..6ae1ef67161d651368535d85c39290de57654398 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -1,4 +1,6 @@
 \documentclass{article}
 \documentclass{article}

+%\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+%\documentclass[preprint,12pt]{elsarticle}

 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}


@@ -7,9 +9,16 @@
 \usepackage{amscd}
 \usepackage{moreverb}
 \usepackage{commath}
 \usepackage{amscd}
 \usepackage{moreverb}
 \usepackage{commath}

-\usepackage{algorithm2e}
+\usepackage[ruled,vlined]{algorithm2e}

 \usepackage{listings}
 \usepackage[standard]{ntheorem}
 \usepackage{listings}
 \usepackage[standard]{ntheorem}

+\usepackage{algorithmic}
+\usepackage{slashbox}
+\usepackage{ctable}
+\usepackage{tabularx}
+\usepackage{multirow}
+
+\usepackage{color}

 
 % Pour mathds : les ensembles IR, IN, etc.
 \usepackage{dsfont}
 
 % Pour mathds : les ensembles IR, IN, etc.
 \usepackage{dsfont}


@@ -21,7 +30,6 @@
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 

-\usepackage{color}

 
 \newtheorem{notation}{Notation}
 
 
 \newtheorem{notation}{Notation}
 


@@ -34,31 +42,57 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 

-\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}

 \begin{document}
 
 \begin{document}
 

-\author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
-Guyeux, and Pierre-Cyrille Heam\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
 
 \maketitle
 

+
+%\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
 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
 \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.
 cards.

-It is finally established that, under reasonable assumptions, the proposed PRNG can be cryptographically 
+It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically 

 secure.
 secure.

+A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is finally proposed.

 
 
 \end{abstract}
 
 
 \end{abstract}

+%}
+%\begin{keyword}
+%   pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
+%\end{keyword}
+%\end{frontmatter}
+
+
+%\IEEEdisplaynotcompsoctitleabstractindextext
+%\IEEEpeerreviewmaketitle
+

 
 \section{Introduction}
 
 
 \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
 ``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


@@ -66,37 +100,43 @@ 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. 
 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,
 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.
 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
+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
 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
+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.
 third requirement, that is to define chaotic generators.

-The main idea is to take benefits from a chaotic dynamical system to obtain a
+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.
 generator that is unpredictable, disordered, sensible to its seed, or in other words chaotic.

-Their desire is to map a given chaotic dynamics into a sequence that seems random 
+Their goal is to map a given chaotic dynamics into a sequence that seems random 

 and unassailable due to chaos.
 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
 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.
 
 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.
 between chaos and security as it is understood in cryptography.
 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
 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,
 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,


@@ -109,10 +149,17 @@ 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.
 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}.
+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
 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
 
 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


@@ -124,46 +171,77 @@ The proposition of this paper is to improve widely the speed of the formerly
 proposed generator, without any lack of chaos or statistical properties.
 In particular, a version of this PRNG on graphics processing units (GPU)
 is proposed.
 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
 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
 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.
-Last, but not least, we show that the proposed post-treatment preserves the
-cryptographical security of the inputted PRNG, when this last has such a 
+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 the latter has such a 

 property.
 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 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. 

-Proofs 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   the  GPU   implementation. 
+The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.
+%Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and a new efficient 
+%implementation on CPU.
+ Section~\ref{sec:efficient PRNG} %{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
 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.
 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
 Such a proof leads to the proposition of a cryptographically secure and

-chaotic generator on GPU based on the famous Blum Blum Shum
-in Section~\ref{sec:CSGPU}.
+chaotic generator on GPU based on the famous Blum Blum Shub
+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.
 
 
 
 
 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}
 
 \label{section:related works}
 

-Numerous research works on defining GPU based PRNGs have yet been proposed  in the
-literature, so that completeness is impossible.
-This is why authors of this document only give reference to the most significant attempts 
+Numerous research works on defining GPU based PRNGs have already been proposed  in the
+literature, so that exhaustivity is impossible.
+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. 
 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. 


@@ -171,7 +249,7 @@ A million numbers  per second will be simply written as
 1MSample/s whereas a billion numbers per second is 1GSample/s.
 
 In \cite{Pang:2008:cec}  a PRNG based on  cellular automata is defined
 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
 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


@@ -180,41 +258,44 @@ 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
 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.
 
 
 Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
 PRNGs on  different computing architectures: CPU,  field-programmable gate array
 CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. 
 However the evaluations of the proposed PRNGs are only statistical ones.
 
 
 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. 
 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 can 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
 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
 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. 
 
 Lastly, Cuda  has developed  a  library for  the  generation of  pseudorandom numbers  called
 Curand~\cite{curand11}.        Several       PRNGs        are       implemented, among
 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
 \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 ever 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
 
 \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.
-\subsection{Devaney's Chaotic Dynamical Systems}
+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}
+\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
 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


@@ -225,7 +306,7 @@ Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f :
 \mathcal{X} \rightarrow \mathcal{X}$.
 
 \begin{definition}
 \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}
 $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
 \varnothing$.
 \end{definition}


@@ -244,7 +325,7 @@ necessarily the same period).
 
 
 \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
 
 
 \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}
 
 topologically transitive.
 \end{definition}
 


@@ -252,12 +333,12 @@ The chaos property is strongly linked to the notion of ``sensitivity'', defined
 on a metric space $(\mathcal{X},d)$ by:
 
 \begin{definition}
 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 $.
 
 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
 \end{definition}
 
 Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is


@@ -281,7 +362,7 @@ Let us consider  a \emph{system} with a finite  number $\mathsf{N} \in
 \mathds{N}^*$ 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
 \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 $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$
 
 \rrbracket $ is called a \emph{strategy}. The set of all strategies is
 denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$
 


@@ -316,15 +397,15 @@ Let us now recall how to define a suitable metric space where chaotic iterations
 are continuous. For further explanations, see, e.g., \cite{guyeux10}.
 
 Let $\delta $ be the \emph{discrete Boolean metric}, $\delta
 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:
-\begin{equation}
+(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function
+$F_{f}:  \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} 
+\longrightarrow  \mathds{B}^{\mathsf{N}}$
+\begin{equation*}

 \begin{array}{lrll}
 \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,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ f(E)_{k}.\overline{\delta
+(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%

 \end{array}%
 \end{array}%

-\end{equation}%
+\end{equation*}%

 \noindent where + and . are the Boolean addition and product operations.
 Consider the phase space:
 \begin{equation}
 \noindent where + and . are the Boolean addition and product operations.
 Consider the phase space:
 \begin{equation}


@@ -380,9 +461,9 @@ 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
 \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 $E, \check{E}$
 \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 $E, \check{E}$


@@ -391,7 +472,7 @@ 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.
 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,
 
 
 Finally, it has been established in \cite{guyeux10} that,
 


@@ -401,7 +482,7 @@ the metric space $(\mathcal{X},d)$.
 \end{proposition}
 
 The chaotic property of $G_f$ has been firstly established for the vectorial
 \end{proposition}
 
 The chaotic property of $G_f$ has been firstly established for the vectorial

-Boolean negation $f(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
+Boolean negation $f_0(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly

 introduced the notion of asynchronous iteration graph recalled bellow.
 
 Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
 introduced the notion of asynchronous iteration graph recalled bellow.
 
 Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The


@@ -414,8 +495,7 @@ The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
 strategy $s$ such that the parallel iteration of $G_f$ from the
 initial point $(s,x)$ reaches the point $x'$.
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
 strategy $s$ such that the parallel iteration of $G_f$ from the
 initial point $(s,x)$ reaches the point $x'$.

-
-We have finally proven in \cite{bcgr11:ip} that,
+We have then proven in \cite{bcgr11:ip} that,

 
 
 \begin{theorem}
 
 
 \begin{theorem}


@@ -424,14 +504,33 @@ Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic  (accord
 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 

-This result of chaos has lead us to study the possibility to build a
+Finally, we have established in \cite{bcgr11:ip} that,
+\begin{theorem}
+  Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
+  iteration graph, $\check{M}$ its adjacency
+  matrix and $M$
+  a $n\times n$ matrix defined by 
+  $
+  M_{ij} = \frac{1}{n}\check{M}_{ij}$ %\textrm{ 
+  if $i \neq j$ and  
+  $M_{ii} = 1 - \frac{1}{n} \sum\limits_{j=1, j\neq i}^n \check{M}_{ij}$ otherwise.
+  
+  If $\Gamma(f)$ is strongly connected, then 
+  the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows 
+  a law that tends to the uniform distribution 
+  if and only if $M$ is a double stochastic matrix.
+\end{theorem} 
+
+
+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}} 
 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 $G_f$
 \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$

-during implementations (due to the discrete nature of $f$). It is as if
+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).
 $\mathds{B}^\mathsf{N}$ 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 both theorems above.

 
 \section{Application to Pseudorandomness}
 \label{sec:pseudorandom}
 
 \section{Application to Pseudorandomness}
 \label{sec:pseudorandom}


@@ -440,30 +539,58 @@ $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracke
 
 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, 
 
 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 improves the statistical properties of each
-generator taken alone. Furthermore, our generator 
-possesses various chaos properties that none of the generators used as input
-present.
+leading thus to a new PRNG that 
+should improve the statistical properties of each
+generator taken alone. 
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as  input present.
+
+

 
 \begin{algorithm}[h!]
 
 \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)}
 $x\leftarrow x^0$\;
 \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
 ($n$ bits)}
 \KwOut{a configuration $x$ ($n$ bits)}
 $x\leftarrow x^0$\;

-$k\leftarrow b + \textit{XORshift}(b)$\;
+$k\leftarrow b + PRNG_1(b)$\;

 \For{$i=0,\dots,k$}
 {
 \For{$i=0,\dots,k$}
 {

-$s\leftarrow{\textit{XORshift}(n)}$\;
+$s\leftarrow{PRNG_2(n)}$\;

 $x\leftarrow{F_f(s,x)}$\;
 }
 return $x$\;
 $x\leftarrow{F_f(s,x)}$\;
 }
 return $x$\;

-%\end{scriptsize}
-\caption{PRNG with chaotic functions}
+\end{small}
+\caption{An arbitrary round of $Old~ CI~ PRNG_f(PRNG_1,PRNG_2)$}

 \label{CI Algorithm}
 \end{algorithm}
 
 \label{CI Algorithm}
 \end{algorithm}
 

+
+
+
+This generator is synthesized in Algorithm~\ref{CI Algorithm}.
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation   des   IC   chaotiques};
+an integer $b$, ensuring that the number of executed iterations
+between two outputs 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
+inputted generators $PRNG_i(k), i=1,2$,
+ which must return integers
+uniformly distributed
+into $\llbracket 1 ; k \rrbracket$.
+For instance, these PRNGs can be the \textit{XORshift}~\cite{Marsaglia2003},
+being a category of very fast PRNGs designed by George Marsaglia
+that repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
+with a bit shifted version of it. Such a PRNG, which has a period of
+$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. 
+This XORshift, or any other reasonable PRNG, is used
+in our own generator to compute both the number of iterations between two
+outputs (provided by $PRNG_1$) and the strategy elements ($PRNG_2$).
+
+%This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.
+
+

 \begin{algorithm}[h!]
 \begin{algorithm}[h!]

+\begin{small}

 \KwIn{the internal configuration $z$ (a 32-bit word)}
 \KwOut{$y$ (a 32-bit word)}
 $z\leftarrow{z\oplus{(z\ll13)}}$\;
 \KwIn{the internal configuration $z$ (a 32-bit word)}
 \KwOut{$y$ (a 32-bit word)}
 $z\leftarrow{z\oplus{(z\ll13)}}$\;


@@ -471,49 +598,114 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\;
 $z\leftarrow{z\oplus{(z\ll5)}}$\;
 $y\leftarrow{z}$\;
 return $y$\;
 $z\leftarrow{z\oplus{(z\ll5)}}$\;
 $y\leftarrow{z}$\;
 return $y$\;

-\medskip
+\end{small}

 \caption{An arbitrary round of \textit{XORshift} algorithm}
 \label{XORshift}
 \end{algorithm}
 
 
 \caption{An arbitrary round of \textit{XORshift} algorithm}
 \label{XORshift}
 \end{algorithm}
 
 

+\subsection{A ``New CI PRNG''}
+
+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 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 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 $\left(S^n\right)_{n\in\mathds{N}}\in 
+\llbracket 1, 32 \rrbracket^\mathds{N}$ is
+an \emph{irregular decimation} of $PRNG_2$ sequence, as described in 
+Algorithm~\ref{Chaotic iteration1}.
+
+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 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 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 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}
+m^n = g(y^n)=
+\left\{
+\begin{array}{l}
+0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\
+1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\
+2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\
+\vdots~~~~~ ~~\vdots~~~ ~~~~\\
+N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
+\end{array}
+\right.
+\end{equation}

 
 

+\begin{algorithm}
+\textbf{Input:} the internal state $x$ (32 bits)\\
+\textbf{Output:} a state $r$ of 32 bits
+\begin{algorithmic}[1]
+\FOR{$i=0,\dots,N$}
+{
+\STATE$d_i\leftarrow{0}$\;
+}
+\ENDFOR
+\STATE$a\leftarrow{PRNG_1()}$\;
+\STATE$k\leftarrow{g(a)}$\;
+\WHILE{$i=0,\dots,k$}
+
+\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
+\STATE$S\leftarrow{b}$\;
+    \IF{$d_S=0$}
+    {
+\STATE      $x_S\leftarrow{ \overline{x_S}}$\;
+\STATE      $d_S\leftarrow{1}$\;
+
+    }
+    \ELSIF{$d_S=1$}
+    {
+\STATE      $k\leftarrow{ k+1}$\;
+    }\ENDIF
+\ENDWHILE\\
+\STATE $r\leftarrow{x}$\;
+\STATE return $r$\;
+\medskip
+\caption{An arbitrary round of the new CI generator}
+\label{Chaotic iteration1}
+\end{algorithmic}
+\end{algorithm}

 
 

-This generator is synthesized in Algorithm~\ref{CI Algorithm}.
-It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation   des   IC   chaotiques};
-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
-uniformly distributed
-into $\llbracket 1 ; k \rrbracket$.
-\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia,
-which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
-with a bit shifted version of it. This PRNG, which has a period of
-$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
-in our PRNG to compute the strategy length and the strategy elements.
-
-
-We have proven in \cite{bcgr11:ip} that,
-\begin{theorem}
-  Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
-  iteration graph, $\check{M}$ its adjacency
-  matrix and $M$ a $n\times n$ matrix defined as in the previous lemma.
-  If $\Gamma(f)$ is strongly connected, then 
-  the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows 
-  a law that tends to the uniform distribution 
-  if and only if $M$ is a double stochastic matrix.
-\end{theorem} 

 
 

-This former generator as successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
+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}
 
 
 \subsection{Improving the Speed of the Former Generator}
 

-Instead of updating only one cell at each iteration, we can try to choose a
-subset of components and to update them together. Such an attempt leads
-to a kind of merger of the two sequences used in Algorithm 
-\ref{CI Algorithm}. When the updating function is the vectorial negation,
+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:
 
 \begin{equation}
 this algorithm can be rewritten as follows:
 
 \begin{equation}


@@ -526,12 +718,17 @@ 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. 
 \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 
 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.
 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 
 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 Oplus} is of 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
 
 The single basic component presented in Eq.~\ref{equation Oplus} is of 
 ordinary use as a good elementary brick in various PRNGs. It corresponds


@@ -550,14 +747,16 @@ 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
 $\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ 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.
 
 
 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
-the vectorial negation, leads to a speed improvement. However, proofs
+Obviously, replacing the previous CI PRNG Algorithms by 
+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).
+However, proofs

 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
 \ref{Def:chaotic iterations}. The question is now to determine whether the
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
 \ref{Def:chaotic iterations}. The question is now to determine whether the


@@ -567,11 +766,11 @@ faster, does not deflate their topological chaos properties.
 \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
 \label{deuxième def}
 Let us consider the discrete dynamical systems in chaotic iterations having 
 \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
 \label{deuxième def}
 Let us consider the discrete dynamical systems in chaotic iterations having 

-the general form:
+the general form: $\forall    n\in     \mathds{N}^{\ast     }$, $  \forall     i\in
+\llbracket1;\mathsf{N}\rrbracket $,

 
 \begin{equation}
 
 \begin{equation}

-\forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
-\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+  x_i^n=\left\{

 \begin{array}{ll}
   x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
   \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
 \begin{array}{ll}
   x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
   \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.


@@ -596,14 +795,13 @@ 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:
 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:

-\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},%
+$F_{f}:  \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} 
+\longrightarrow \mathds{B}^{\mathsf{N}}$
+\begin{equation*}
+\begin{array}{rll}
+ (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%

 \end{array}%
 \end{array}%

-\end{equation}%
+\end{equation*}%

 where + and . are the Boolean addition and product operations, and $\overline{x}$ 
 is the negation of the Boolean $x$.
 Consider the phase space:
 where + and . are the Boolean addition and product operations, and $\overline{x}$ 
 is the negation of the Boolean $x$.
 Consider the phase space:


@@ -613,7 +811,7 @@ Consider the phase space:
 \end{equation}
 \noindent and the map defined on $\mathcal{X}$:
 \begin{equation}
 \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
 \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


@@ -630,7 +828,7 @@ X^{k+1}=G_{f}(X^k).%
 \right.
 \end{equation}%
 
 \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 
 iterations. 
 
 To study the Devaney's chaos property, a distance between two points 


@@ -640,17 +838,21 @@ Let us introduce:
 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
 \label{nouveau d}
 \end{equation}
 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
 $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
 
 where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
 $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
 


@@ -661,7 +863,7 @@ 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
 
 \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 $d$, 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 
  \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 


@@ -678,7 +880,7 @@ inequality is obtained.
 
 
 Before being able to study the topological behavior of the general 
 
 
 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 
 
 \begin{proposition}
  For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 


@@ -714,7 +916,7 @@ 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}
 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
 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


@@ -729,14 +931,18 @@ thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
 \noindent As a consequence, the $k+1$ first entries of the strategies of $%
 G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
 the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
 \noindent As a consequence, the $k+1$ first entries of the strategies of $%
 G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
 the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%

-10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline
+10^{-(k+1)}\leqslant \varepsilon $.
+

 In conclusion,
 In conclusion,

-$$
-\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)
+%%RAPH : ici j'ai rajouté une ligne
+%%TOF : ici j'ai rajouté un commentaire
+%%TOF : ici aussi
+$
+\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+,$ $\forall n\geqslant N_{0},$
+$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)

 \leqslant \varepsilon .
 \leqslant \varepsilon .

-$$
+$

 $G_{f}$ is consequently continuous.
 \end{proof}
 
 $G_{f}$ is consequently continuous.
 \end{proof}
 


@@ -776,7 +982,7 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
 claimed in the lemma.
 \end{proof}
 
 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.
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 Firstly, strong transitivity implies transitivity.


@@ -794,8 +1000,10 @@ 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
 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 $t<t_1$, by the choice of $t_1$, we
 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 $t<t_1$, by the choice of $t_1$, we


@@ -803,51 +1011,381 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \end{proof}
 
 
 \end{proof}
 
 

-
-\section{Efficient PRNG based on Chaotic Iterations}
+%\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) 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, 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}
+
+
+%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}
+%\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}
 
 \label{sec:efficient PRNG}
 

-Based on the proof presented in the previous section, it is now possible to 
-improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. 
-The first idea is to consider
-that the provided strategy is a pseudorandom Boolean vector obtained by a
-given PRNG.
-An iteration of the system is simply the bitwise exclusive or between
-the last computed state and the current strategy.
-Topological properties of disorder exhibited by chaotic 
-iterations can be inherited by the inputted generator, hoping by doing so to 
-obtain some statistical improvements while preserving speed.
-
-
-Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
-are
-done.  
-Suppose  that $x$ and the  strategy $S^i$ are given as
-binary vectors.
-Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
-
-\begin{table}
-$$
-\begin{array}{|cc|cccccccccccccccc|}
-\hline
-x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
-\hline
-S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
-\hline
-x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
-\hline
-
-\hline
- \end{array}
-$$
-\caption{Example of an arbitrary round of the proposed generator}
-\label{TableExemple}
-\end{table}
-
-
-
-\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG}
+\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}. 
+%The first idea is to consider
+%that the provided strategy is a pseudorandom Boolean vector obtained by a
+%given PRNG.
+%An iteration of the system is simply the bitwise exclusive or between
+%the last computed state and the current strategy.
+%Topological properties of disorder exhibited by chaotic 
+%iterations can be inherited by the inputted generator, we hope by doing so to 
+%obtain some statistical improvements while preserving speed.
+%
+%%RAPH : j'ai viré tout ca
+%% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
+%% are
+%% done.  
+%% Suppose  that $x$ and the  strategy $S^i$ are given as
+%% binary vectors.
+%% Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
+
+%% \begin{table}
+%% \begin{scriptsize}
+%% $$
+%% \begin{array}{|cc|cccccccccccccccc|}
+%% \hline
+%% x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+%% \hline
+%% S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+%% \hline
+%% x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+%% \hline
+
+%% \hline
+%%  \end{array}
+%% $$
+%% \end{scriptsize}
+%% \caption{Example of an arbitrary round of the proposed generator}
+%% \label{TableExemple}
+%% \end{table}
+
+
+
+
+\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}}
+\begin{small}

 \begin{lstlisting}
 \begin{lstlisting}

+

 unsigned int CIPRNG() {
   static unsigned int x = 123123123;
   unsigned long t1 = xorshift();
 unsigned int CIPRNG() {
   static unsigned int x = 123123123;
   unsigned long t1 = xorshift();


@@ -862,54 +1400,59 @@ unsigned int CIPRNG() {
   return x;
 }
 \end{lstlisting}
   return x;
 }
 \end{lstlisting}

+\end{small}

 
 
 
 
 
 

+In Listing~\ref{algo:seqCIPRNG} a sequential  version of the proposed PRNG based
+on  chaotic  iterations  is  presented.   The xor  operator  is  represented  by
+\textasciicircum.  This function uses  three classical 64-bits PRNGs, namely the
+\texttt{xorshift},         the          \texttt{xor128},         and         the
+\texttt{xorwow}~\cite{Marsaglia2003}.  In the following, we call them ``xor-like
+PRNGs''.   As each  xor-like PRNG  uses 64-bits  whereas our  proposed generator
+works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the
+32 least  significant bits  of a given  integer, and the  code \texttt{(unsigned
+  int)(t$>>$32)} in order to obtain the 32 most significant bits of \texttt{t}.

 
 

+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}. 
+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.

 
 

-In Listing~\ref{algo:seqCIPRNG}  a sequential version of  the proposed PRNG based on chaotic iterations
- is  presented.  The xor operator is  represented by \textasciicircum.
-This  function uses  three classical  64-bits PRNGs, namely the  \texttt{xorshift}, the
-\texttt{xor128},  and  the  \texttt{xorwow}~\cite{Marsaglia2003}.   In  the following,  we  call  them
-``xor-like PRNGs''. 
-As
-each xor-like PRNG  uses 64-bits whereas our proposed generator works with 32-bits,
-we use the command \texttt{(unsigned int)}, that selects the 32 least significant bits of a given integer, and the code
-\texttt{(unsigned int)(t3$>>$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 that are provided by 3 64-bits PRNGs.   This version successfully passes the
-stringent BigCrush battery of tests~\cite{LEcuyerS07}.

 
 

-\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
 \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
 Obviously, having these requirements in  mind, it is possible to build

-a   program    similar   to    the   one   presented    in   Algorithm
+a   program    similar   to    the   one   presented    in  Listing 

 \ref{algo:seqCIPRNG}, which computes  pseudorandom numbers on GPU.  To
 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
 \ref{algo:seqCIPRNG}, which computes  pseudorandom numbers on GPU.  To
 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. With  CUDA parts of  the code which  are executed by the  GPU are
+them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU, are

 called {\it kernels}.
 
 
 \subsection{Naive Version for GPU}
 
  
 called {\it kernels}.
 
 
 \subsection{Naive Version for GPU}
 
  

-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.  
+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. 
 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.
 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
 parameters embedded into each thread.   
 
 The implementation of  the three


@@ -919,8 +1462,9 @@ 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.
 
 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\;}
 \KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
 PRNGs in global memory\;
 NumThreads: number of threads\;}


@@ -933,14 +1477,16 @@ NumThreads: number of threads\;}
   }
   store internal variables in InternalVarXorLikeArray[threadIdx]\;
 }
   }
   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}
 
 \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
 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)},
 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)},


@@ -948,22 +1494,29 @@ then   the  memory   required   to  store all of the  internals   variables  of
 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.
 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 
 
 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}
 
 \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
 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}
 
 \end{remark}
 
 \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
 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


@@ -972,45 +1525,47 @@ thread uses the result of which other  one, we can use a combination array that
 contains  the indexes  of  all threads  and  for which  a combination has  been
 performed. 
 
 contains  the indexes  of  all threads  and  for which  a combination has  been
 performed. 
 

-In Algorithm~\ref{algo:gpu_kernel2}, two combination arrays are used.
-The    variable   \texttt{offset}    is    computed   using    the   value    of
+In  Algorithm~\ref{algo:gpu_kernel2},  two  combination  arrays are  used.   The
+variable     \texttt{offset}    is     computed    using     the     value    of

 \texttt{combination\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}
 \texttt{combination\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}

-representing the indexes of the  other threads whose results are used
-by the  current one. In  this algorithm, we  consider that a  64-bits xor-like
-PRNG has been chosen, and so its two 32-bits parts are used.
+representing the  indexes of  the other  threads whose results  are used  by the
+current one.   In this algorithm, we  consider that a 32-bits  xor-like PRNG has
+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{algorithm}

-
+\begin{small}

 \KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
 in global memory\;
 NumThreads: Number of threads\;
 \KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
 in global memory\;
 NumThreads: Number of threads\;

-tab1, tab2: Arrays containing combinations of size combination\_size\;}
+array\_comb1, array\_comb2: Arrays containing combinations of size combination\_size\;}

 
 \KwOut{NewNb: array containing random numbers in global memory}
 
 \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\;
   offset = threadIdx\%combination\_size\;

-  o1 = threadIdx-offset+tab1[offset]\;
-  o2 = threadIdx-offset+tab2[offset]\;
+  o1 = threadIdx-offset+array\_comb1[offset]\;
+  o2 = threadIdx-offset+array\_comb2[offset]\;

   \For{i=1 to n} {
     t=xor-like()\;
   \For{i=1 to n} {
     t=xor-like()\;

-    t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\;
-    shared\_mem[threadId]=t\;
-    x = x $\hat{ }$ t\;
+    t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;
+    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]\;

 }
 }

-
-\caption{main kernel for the chaotic iterations based PRNG GPU efficient
-version}
-\label{algo:gpu_kernel2}
+\end{small}
+\caption{Main kernel for the chaotic iterations based PRNG GPU efficient
+version\label{IR}}
+\label{algo:gpu_kernel2} 

 \end{algorithm}
 
 \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
 
 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


@@ -1021,8 +1576,8 @@ 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 
 $\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
 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 $\mathds{B}^ \mathsf{N}$.
-To prevent from any flaws of chaotic properties, we must check that the right 
+The left term $x$ obviously belongs to $\mathds{B}^ \mathsf{N}$.
+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 $\llbracket 1, \mathsf{N} \rrbracket$. 
 
 term (the last $t$), corresponding to the strategies,  can possibly be equal to any
 integer of $\llbracket 1, \mathsf{N} \rrbracket$. 
 


@@ -1032,8 +1587,7 @@ last $t$ respects the requirement. Furthermore, it is possible to
 prove by an immediate mathematical induction that, as the initial $x$
 is uniformly distributed (it is provided by a cryptographically secure PRNG),
 the two other stored values shmem[o1] and shmem[o2] are uniformly distributed too,
 prove by an immediate mathematical induction that, as the initial $x$
 is uniformly distributed (it is provided by a cryptographically secure PRNG),
 the two other stored values shmem[o1] and shmem[o2] are uniformly distributed too,

-(this can be stated by an immediate mathematical
-induction), and thus the next $x$ is finally uniformly distributed.
+(this is the induction hypothesis), and thus the next $x$ is finally uniformly distributed.

 
 Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
 chaotic iterations presented previously, and for this reason, it satisfies the 
 
 Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
 chaotic iterations presented previously, and for this reason, it satisfies the 


@@ -1051,7 +1605,7 @@ All the
 cards have 240 cores.
 
 In  Figure~\ref{fig:time_xorlike_gpu} we  compare the  quantity of  pseudorandom numbers
 cards have 240 cores.
 
 In  Figure~\ref{fig:time_xorlike_gpu} we  compare the  quantity of  pseudorandom numbers

-generated per second with various xor-like based PRNG. In this figure, the optimized
+generated per second with various xor-like based PRNGs. In this figure, the optimized

 versions use the {\it xor64} described in~\cite{Marsaglia2003}, whereas the naive versions
 embed  the three  xor-like  PRNGs described  in Listing~\ref{algo:seqCIPRNG}.   In
 order to obtain the optimal performances, the storage of pseudorandom numbers
 versions use the {\it xor64} described in~\cite{Marsaglia2003}, whereas the naive versions
 embed  the three  xor-like  PRNGs described  in Listing~\ref{algo:seqCIPRNG}.   In
 order to obtain the optimal performances, the storage of pseudorandom numbers


@@ -1059,18 +1613,18 @@ 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
 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
 practice,  the Tesla C1060  has more  memory than  the GTX  280, and  this memory
 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
 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}
 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[scale=.7]{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}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
 \label{fig:time_xorlike_gpu}


@@ -1080,17 +1634,16 @@ As a  comparison,   Listing~\ref{algo:seqCIPRNG}  leads   to the  generation of
 
 
 
 
 
 

-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
-reduction. 
+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  necessarily paid by  a speed
+reduction.

 
 \begin{figure}[htbp]
 \begin{center}
 
 \begin{figure}[htbp]
 \begin{center}

-  \includegraphics[scale=.7]{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}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
 \label{fig:time_bbs_gpu}


@@ -1098,10 +1651,10 @@ 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.
 
 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,
 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.

 
 
 
 
 
 


@@ -1110,34 +1663,52 @@ as it is shown in the next sections.
 
 
 \section{Security Analysis}
 
 
 \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 $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any
 
 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 $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
-$\ell_G(N)$ with $\ell_G(N)>N$.
+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
 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,
-$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(N)},$$
+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
 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}
 
 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 $\ell^\prime(N)>N)$~\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,
 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,


@@ -1148,7 +1719,7 @@ 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
 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.
 
 We claim now that if this PRNG is secure,
 then the new one is secure too.
 


@@ -1159,7 +1730,7 @@ PRNG too.
 \end{proposition}
 
 \begin{proof}
 \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 
 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 


@@ -1192,8 +1763,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. 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
 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}
 \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 $x$,
 \end{equation}
 
 Now, using (\ref{PCH-1}) again, one has  for every $x$,


@@ -1202,7 +1775,7 @@ 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
 \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. 
 D^\prime(H(x))=D(yx),
 \end{equation}
 where $y$ is randomly generated. 


@@ -1212,14 +1785,99 @@ 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
 \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 
 $$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$
 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}
 
 
 \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}
 \section{Cryptographical Applications}
 
 \subsection{A Cryptographically Secure PRNG for GPU}


@@ -1229,110 +1887,164 @@ It is  possible to build a  cryptographically secure PRNG based  on the previous
 algorithm (Algorithm~\ref{algo:gpu_kernel2}).   Due to Proposition~\ref{cryptopreuve},
 it simply consists  in replacing
 the  {\it  xor-like} PRNG  by  a  cryptographically  secure one.  
 algorithm (Algorithm~\ref{algo:gpu_kernel2}).   Due to Proposition~\ref{cryptopreuve},
 it simply consists  in replacing
 the  {\it  xor-like} PRNG  by  a  cryptographically  secure one.  

-We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form:
-$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers. These
-prime numbers  need to be congruent  to 3 modulus  4. BBS is
+We have chosen the Blum Blum Shub generator~\cite{BBS} (usually denoted by BBS) having the form:
+$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers (these
+prime numbers  need to be congruent  to 3 modulus  4). BBS is known to be

 very slow and only usable for cryptographic applications. 
 
   
 The modulus operation is the most time consuming operation for current
 GPU cards.  So in order to obtain quite reasonable performances, it is
 very slow and only usable for cryptographic applications. 
 
   
 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 less than $2^{32}$  and the number $M$  need to be
-less 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
+required to use only modulus  on 32-bits integer numbers. Consequently
+$x_n^2$ 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
 4 least significant bits of $x_n$ can be chosen (the maximum number of
 indistinguishable    bits    is    lesser    than   or    equals    to

-$log_2(log_2(x_n))$). So to generate a  32 bits number, we need to use
-8 times  the BBS  algorithm with different  combinations of  $M$. This
-approach is  not sufficient to pass  all the tests  of TestU01 because
-the fact  of having chosen  small values of  $M$ for the BBS  leads to
-have a  small period. So, in  order to add randomness  we proceed with
+$log_2(log_2(M))$). 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 tests of TestU01,
+as small values of  $M$ for the BBS  lead to
+  small periods. So, in  order to add randomness  we have proceeded with

 the followings  modifications. 
 \begin{itemize}
 \item
 the followings  modifications. 
 \begin{itemize}
 \item

-First we  define 16 arrangement arrays  instead of 2  (as described in
-algorithm \ref{algo:gpu_kernel2}) but only 2  are used at each call of
-the  PRNG kernels. In  practice, the  selection of  which combinations
-arrays will be used is different for all the threads and is determined
+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   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.
 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  \& performs the  AND bitwise.  So using  \&7 with  a number
-gives the last 3 bits, so it provides a number between 0 and 7.
+%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, thus providing a number between 0 and 7.

 \item
 \item

-Second, after the  generation of the 8 BBS numbers  for each thread we
-have a 32 bits number for which the period is possibly quite small. So
-to add randomness,  we generate 4 more BBS numbers  which allows us to
-shift  the 32 bits  numbers and  add upto  6 new  bits.  This  part is
-described  in algorithm~\ref{algo:bbs_gpu}.  In  practice, if  we call
-{\it strategy}, the number representing  the strategy, the last 2 bits
-of the first new BBS number are  used to make a left shift of at least
-3 bits. The  last 3 bits of the  second new BBS number are  add to the
+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
+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  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.
 \item
 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.
 \item

-Finally, as  we use 8 BBS numbers  for each thread, the  store of these
+Finally, as  we use 8 BBS numbers  for each thread, the  storage of these

 numbers at the end of the  kernel is performed using a rotation. So,
 internal  variable for  BBS number  1 is  stored in  place  2, internal
 numbers at the end of the  kernel is performed using a rotation. So,
 internal  variable for  BBS number  1 is  stored in  place  2, internal

-variable  for BBS  number 2  is  store ind  place 3,  ... and  internal
+variable  for BBS  number 2  is  stored in  place 3,  ..., and finally, internal

 variable for BBS number 8 is stored in place 1.
 \end{itemize}
 
 variable for BBS number 8 is stored in place 1.
 \end{itemize}
 

-

 \begin{algorithm}
 \begin{algorithm}

-
+\begin{small}

 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 in global memory\;
 NumThreads: Number of threads\;
 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 in global memory\;
 NumThreads: Number of threads\;

-tab: 2D Arrays containing 16 combinations (in first dimension)  of size combination\_size (in second dimension)\;}
+array\_comb: 2D Arrays containing 16 combinations (in first dimension)  of size combination\_size (in second dimension)\;
+array\_shift[4]=\{0,1,3,7\}\;
+}

 
 \KwOut{NewNb: array containing random numbers in global memory}
 
 \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\;
   we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\;
   offset = threadIdx\%combination\_size\;

-  o1 = threadIdx-offset+tab[bbs1\&7][offset]\;
-  o2 = threadIdx-offset+tab[8+bbs2\&7][offset]\;
+  o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
+  o2 = threadIdx-offset+array\_comb[8+bbs2\&7][offset]\;

   \For{i=1 to n} {
   \For{i=1 to n} {

-    t<<=4\;
+    t$<<$=4\;

     t|=BBS1(bbs1)\&15\;
     ...\;
     t|=BBS1(bbs1)\&15\;
     ...\;

-    t<<=4\;
+    t$<<$=4\;

     t|=BBS8(bbs8)\&15\;
     t|=BBS8(bbs8)\&15\;

-    //two new shifts\;
-    t<<=BBS3(bbs3)\&3\;
-    t|=BBS1(bbs1)\&7\;
-     t<<=BBS7(bbs7)\&3\;
-    t|=BBS2(bbs2)\&7\;
-    t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\;
-    shared\_mem[threadId]=t\;
-    x = x $\hat{ }$ t\;
-
-    store the new PRNG in NewNb[NumThreads*threadId+i]\;
+    \tcp{two new shifts}
+    shift=BBS3(bbs3)\&3\;
+    t$<<$=shift\;
+    t|=BBS1(bbs1)\&array\_shift[shift]\;
+    shift=BBS7(bbs7)\&3\;
+    t$<<$=shift\;
+    t|=BBS2(bbs2)\&array\_shift[shift]\;
+    t=t\textasciicircum  shmem[o1]\textasciicircum     shmem[o2]\;
+    shared\_mem[threadIdx]=t\;
+    x = x\textasciicircum   t\;
+
+    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}
 \label{algo:bbs_gpu}
 \end{algorithm}
 
 \caption{main kernel for the BBS based PRNG GPU}
 \label{algo:bbs_gpu}
 \end{algorithm}
 

-In algorithm~\ref{algo:bbs_gpu}, t<<=4 performs a left shift of 4 bits
-on the variable  t and stores the result  in t. BBS1(bbs1)\&15 selects
-the last  four bits of the result  of BBS1. It should  be noticed that
-for the two new shifts, we use arbitrarily 4 BBSs that have previously
-been used.
-
-
-
-\subsection{A Cryptographically Secure and Chaotic Asymetric Cryptosystem}
+In Algorithm~\ref{algo:bbs_gpu}, $n$ is for  the quantity of random numbers that
+a thread has to  generate.  The operation t<<=4 performs a left  shift of 4 bits
+on the variable  $t$ and stores the result in  $t$, and $BBS1(bbs1)\&15$ selects
+the last  four bits  of the  result of $BBS1$.   Thus an  operation of  the form
+$t<<=4; t|=BBS1(bbs1)\&15\;$  realizes in $t$ a  left shift of 4  bits, and then
+puts the 4 last bits of $BBS1(bbs1)$  in the four last positions of $t$.  Let us
+remark that the initialization $t$ is not a  necessity as we fill it 4 bits by 4
+bits, until  having obtained 32-bits.  The  two last new shifts  are realized in
+order to enlarge the small periods of  the BBS used here, to introduce a kind of
+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
+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).
+
+It should  be noticed that this generator has once more the form $x^{n+1} = x^n \oplus S^n$,
+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.
+
+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.
+
+
+\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
+\label{Blum-Goldwasser}
+We finish this research work by giving some thoughts about the use of
+the proposed PRNG in an asymmetric cryptosystem.
+This first approach will be further investigated in a future work.

 
 \subsubsection{Recalls of the Blum-Goldwasser Probabilistic Cryptosystem}
 
 The Blum-Goldwasser cryptosystem is a cryptographically secure asymmetric key encryption algorithm 
 proposed in 1984~\cite{Blum:1985:EPP:19478.19501}.  The encryption algorithm 
 
 \subsubsection{Recalls of the Blum-Goldwasser Probabilistic Cryptosystem}
 
 The Blum-Goldwasser cryptosystem is a cryptographically secure asymmetric key encryption algorithm 
 proposed in 1984~\cite{Blum:1985:EPP:19478.19501}.  The encryption algorithm 

-implements an XOR-based stream cipher using the BBS PRNG, in order to generate 
+implements a XOR-based stream cipher using the BBS PRNG, in order to generate 

 the keystream. Decryption is done by obtaining the initial seed thanks to
 the final state of the BBS generator and the secret key, thus leading to the
  reconstruction of the keystream.
 the keystream. Decryption is done by obtaining the initial seed thanks to
 the final state of the BBS generator and the secret key, thus leading to the
  reconstruction of the keystream.


@@ -1345,48 +2057,81 @@ The public key is $N$, whereas the secret key is the factorization $(p,q)$.
 
 Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Alice:
 \begin{enumerate}
 
 Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Alice:
 \begin{enumerate}

-\item Bob picks an integer $r$ randomly in the interval $[1,N$ and computes $x_0 = r^2~mod~N$.
+\item Bob picks an integer $r$ randomly in the interval $\llbracket 1,N\rrbracket$ and computes $x_0 = r^2~mod~N$.

 \item He uses the BBS to generate the keystream of $L$ pseudorandom bits $(b_0, \dots, b_{L-1})$, as follows. For $i=0$ to $L-1$,
 \begin{itemize}
 \item $i=0$.
 \item While $i \leqslant L-1$:
 \begin{itemize}
 \item He uses the BBS to generate the keystream of $L$ pseudorandom bits $(b_0, \dots, b_{L-1})$, as follows. For $i=0$ to $L-1$,
 \begin{itemize}
 \item $i=0$.
 \item While $i \leqslant L-1$:
 \begin{itemize}

-\item Set $b_i$ equal to the least-significant\footnote{BBS can securely output up to O(loglogN) of the least-significant bits of xi during each round.} bit of $x_i$,
+\item Set $b_i$ equal to the least-significant\footnote{As signaled previously, BBS can securely output up to $\mathsf{N} = \lfloor log(log(N)) \rfloor$ of the least-significant bits of $x_i$ during each round.} bit of $x_i$,

 \item $i=i+1$,
 \item $x_i = (x_{i-1})^2~mod~N.$
 \end{itemize}
 \end{itemize}
 \item $i=i+1$,
 \item $x_i = (x_{i-1})^2~mod~N.$
 \end{itemize}
 \end{itemize}

-\item The ciphertext is computed by XORing the plaintext bits $m$ with the keystream: $ c = (c_0, \dots, c_{L-1}) = m \oplus  b$.
+\item The ciphertext is computed by XORing the plaintext bits $m$ with the keystream: $ c = (c_0, \dots, c_{L-1}) = m \oplus  b$. This ciphertext is $[c, y]$, where $y=x_{0}^{2^{L}}~mod~N.$

 \end{enumerate}
 \end{enumerate}

-The ciphertext is $(c, y)$, where $y=x_{0}^{2^{L}}~mod~N.$.

 
 
 
 

-When Alice receives $(c_0, \dots, c_{L-1}), y$, she can recover $m$ as follows:
+When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ as follows:

 \begin{enumerate}
 \item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
 \begin{enumerate}
 \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 Recompute the bit-vector $b$ by using BBS and $x_0$.
-\item Compute finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.
+\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 finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.

 \end{enumerate}
 
 
 \subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser}
 
 \end{enumerate}
 
 
 \subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser}
 

+We propose to adapt the Blum-Goldwasser protocol as follows. 
+Let $\mathsf{N} = \lfloor log(log(N)) \rfloor$ be the number of bits that can
+be obtained securely with the BBS generator using the public key $N$ of Alice.
+Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too, and
+her new public key will be $(S^0, N)$.

 
 

+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 $$\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right).$$ 

 
 

-\section{Conclusion}
-
+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 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.

 
 

-In  this  paper  we have  presented  a  new  class  of  PRNGs based  on  chaotic
-iterations. We have proven that these PRNGs are chaotic in the sense of Devaney.
-We also propose a PRNG cryptographically secure and its implementation on GPU.
+\section{Conclusion}

 
 

-An  efficient implementation  on  GPU based  on  a xor-like  PRNG  allows us  to
-generate   a  huge   number   of  pseudorandom   numbers   per  second   (about
-20Gsamples/s). This PRNG succeeds to pass the hardest batteries of TestU01.

 
 

-In future  work we plan to  extend this work  for parallel PRNG for  clusters or
-grid computing.
+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
+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
+secure, then it is the case too for the PRNG we propose, thus leading to
+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.
+\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.