truc bidule

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index c646316ef6549811294655cd3dbb38366f54663f..966dbaaf649a530ab15e9709710bab56a9ae11f8 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -1,4 +1,5 @@
-\documentclass{article}
+%\documentclass{article}
+\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}

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


@@ -7,9 +8,11 @@
 \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}

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


@@ -38,17 +41,17 @@
 \begin{document}
 
 \author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
 \begin{document}
 
 \author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe

-Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}
+Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}

    
    

-\maketitle

 
 

+\IEEEcompsoctitleabstractindextext{

 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
 graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations.  It
 is firstly proven  to be chaotic according to the Devaney's  formulation. We thus propose  an efficient
 implementation  for  GPU that successfully passes the   {\it BigCrush} tests, deemed to be the  hardest
 battery of tests in TestU01.  Experiments show that this PRNG can generate
 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
 graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations.  It
 is firstly proven  to be chaotic according to the Devaney's  formulation. We thus propose  an efficient
 implementation  for  GPU that successfully passes the   {\it BigCrush} tests, deemed to be the  hardest
 battery of tests in TestU01.  Experiments show that this PRNG can generate

-about 20 billions of random numbers  per second on Tesla C1060 and NVidia GTX280
+about 20 billion of random numbers  per second on Tesla C1060 and NVidia GTX280

 cards.
 It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically 
 secure.
 cards.
 It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically 
 secure.


@@ -56,10 +59,17 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin
 
 
 \end{abstract}
 
 
 \end{abstract}

+}
+
+\maketitle
+
+\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


@@ -67,18 +77,18 @@ generator (TRNG).
 In this paper, we focus on reproducible generators, useful for instance in
 Monte-Carlo based simulators or in several cryptographic schemes.
 These domains need PRNGs that are statistically irreproachable. 
 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.
 On the other side, speed is not the main requirement in cryptography: the great
 reported, when the parallelization of a good PRNG is realized.
 This is why ad-hoc PRNGs for each possible architecture must be found to
 achieve both speed and randomness.
 On the other side, speed is not the main requirement in cryptography: the great

-need is to define \emph{secure} generators being able to withstand malicious
+need is to define \emph{secure} generators able to withstand malicious

 attacks. Roughly speaking, an attacker should not be able in practice to make 
 the distinction between numbers obtained with the secure generator and a true random
 sequence. 
 attacks. Roughly speaking, an attacker should not be able in practice to make 
 the distinction between numbers obtained with the secure generator and a true random
 sequence. 

-Finally, a small part of the community working in this domain focus on a
+Finally, a small part of the community working in this domain focuses on a

 third requirement, that is to define chaotic generators.
 The main idea is to take benefits from a chaotic dynamical system to obtain a
 generator that is unpredictable, disordered, sensible to its seed, or in other word chaotic.
 third requirement, that is to define chaotic generators.
 The main idea is to take benefits from a chaotic dynamical system to obtain a
 generator that is unpredictable, disordered, sensible to its seed, or in other word chaotic.


@@ -95,7 +105,7 @@ This is why the use of chaos for PRNG still remains marginal and disputable.
 The authors' opinion is that topological properties of disorder, as they are
 properly defined in the mathematical theory of chaos, can reinforce the quality
 of a PRNG. But they are not substitutable for security or statistical perfection.
 The authors' opinion is that topological properties of disorder, as they are
 properly defined in the mathematical theory of chaos, can reinforce the quality
 of a PRNG. But they are not substitutable for security or statistical perfection.

-Indeed, to the authors' point of view, such properties can be useful in the two following situations. On the
+Indeed, to the authors' mind, such properties can be useful in the two following situations. On the

 one hand, a post-treatment based on a chaotic dynamical system can be applied
 to a PRNG statistically deflective, in order to improve its statistical 
 properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.
 one hand, a post-treatment based on a chaotic dynamical system can be applied
 to a PRNG statistically deflective, in order to improve its statistical 
 properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.


@@ -110,7 +120,7 @@ Let us finish this paragraph by noticing that, in this paper,
 statistical perfection refers to the ability to pass the whole 
 {\it BigCrush} battery of tests, which is widely considered as the most
 stringent statistical evaluation of a sequence claimed as random.
 statistical perfection refers to the ability to pass the whole 
 {\it BigCrush} battery of tests, which is widely considered as the most
 stringent statistical evaluation of a sequence claimed as random.

-This battery can be found into the well-known TestU01 package~\cite{LEcuyerS07}.
+This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}.

 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 
 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 


@@ -131,11 +141,11 @@ applications. Therefore,  it is important  to be able to  generate pseudorandom
 numbers inside a GPU when a scientific application runs in it. This remark
 motivates our proposal of a chaotic and statistically perfect PRNG for GPU.  
 Such device
 numbers inside a GPU when a scientific application runs in it. This remark
 motivates our proposal of a chaotic and statistically perfect PRNG for GPU.  
 Such device

-allows us to generated almost 20 billions of pseudorandom numbers per second.
+allows us to generate almost 20 billion of pseudorandom numbers per second.

 Furthermore, we show that the proposed post-treatment preserves the
 cryptographical security of the inputted PRNG, when this last has such a 
 property.
 Furthermore, we show that the proposed post-treatment preserves the
 cryptographical security of the inputted PRNG, when this last has such a 
 property.

-Last, but not least, we propose a rewritten of the Blum-Goldwasser asymmetric
+Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric

 key encryption protocol by using the proposed method.
 
 The remainder of this paper  is organized as follows. In Section~\ref{section:related
 key encryption protocol by using the proposed method.
 
 The remainder of this paper  is organized as follows. In Section~\ref{section:related


@@ -153,7 +163,7 @@ 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 provided by the post-treatment.
 Such a proof leads to the proposition of a cryptographically secure and
 generator is cryptographically secure, then it is the case too for the
 generator provided by the post-treatment.
 Such a proof leads to the proposition of a cryptographically secure and

-chaotic generator on GPU based on the famous Blum Blum Shum
+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
 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


@@ -165,8 +175,8 @@ summarized and intended future work is presented.
 \section{Related works on GPU based PRNGs}
 \label{section:related works}
 
 \section{Related works on GPU based PRNGs}
 \label{section:related works}
 

-Numerous research works on defining GPU based PRNGs have yet been proposed  in the
-literature, so that completeness is impossible.
+Numerous research works on defining GPU based PRNGs have already been proposed  in the
+literature, so that exhaustivity is impossible.

 This is why authors of this document only give reference to the most significant attempts 
 in this domain, from their subjective point of view. 
 The  quantity of pseudorandom numbers generated per second is mentioned here 
 This is why authors of this document only give reference to the most significant attempts 
 in this domain, from their subjective point of view. 
 The  quantity of pseudorandom numbers generated per second is mentioned here 


@@ -184,7 +194,7 @@ chaos or cryptography in this document.
 In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
 based on  Lagged Fibonacci or Hybrid  Taus.  They have  used these
 PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
 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.
 
 CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. 
 However the evaluations of the proposed PRNGs are only statistical ones.
 


@@ -200,7 +210,7 @@ However, we notice that authors can ``only'' generate between 11 and 16GSamples/
 with a GTX 280  GPU, which should be compared with
 the results presented in this document.
 We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only
 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
 
 Lastly, Cuda  has developed  a  library for  the  generation of  pseudorandom numbers  called
 Curand~\cite{curand11}.        Several       PRNGs        are       implemented, among


@@ -210,13 +220,16 @@ their  fastest version provides  15GSamples/s on  the new  Fermi C2050  card.
 But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
 \newline
 \newline
 But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
 \newline
 \newline

-We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded.
+We can finally remark that, to the best of our knowledge, no GPU implementation has been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered.

 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of

-topological chaos and chaotic iterations.
+topological chaos and chaotic iterations. We assume the reader is familiar
+with basic notions on topology (see for instance~\cite{Devaney}).
+
+

 \subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 \subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$


@@ -229,7 +242,7 @@ 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}


@@ -248,7 +261,7 @@ 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}
 


@@ -256,12 +269,12 @@ 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


@@ -320,15 +333,15 @@ 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}


@@ -384,9 +397,9 @@ their distance should increase too.
 \item In addition, if two systems present the same cells and their respective
 strategies start with the same terms, then the distance between these two points
 must be small because the evolution of the two systems will be the same for a
 \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 $E, \check{E}$


@@ -395,7 +408,7 @@ measure of the differences between strategies $S$ and $\check{S}$. More
 precisely, this floating part is less than $10^{-k}$ if and only if the first
 $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is
 nonzero, then the $k^{th}$ terms of the two strategies are different.
 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,
 


@@ -405,7 +418,7 @@ 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


@@ -445,15 +458,15 @@ Finally, we have established in \cite{bcgr11:ip} that,
 \end{theorem} 
 
 
 \end{theorem} 
 
 

-These results of chaos and uniform distribution have lead us to study the possibility to build a
+These results of chaos and uniform distribution have led us to study the possibility of building a

 pseudorandom number generator (PRNG) based on the chaotic iterations. 
 As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathds{N}} 
 pseudorandom number generator (PRNG) based on the chaotic iterations. 
 As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathds{N}} 

-\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
+\times \mathds{B}^\mathsf{N}$, is built from Boolean networks $f : \mathds{B}^\mathsf{N}

 \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
 during implementations (due to the discrete nature of $f$). Indeed, it is as if
 $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
 \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).
 \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
 during implementations (due to the discrete nature of $f$). Indeed, it is as if
 $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
 \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).

-Let us finally remark that the vectorial negation satisfies the hypotheses of the two theorems above.
+Let us finally remark that the vectorial negation satisfies the hypotheses of both theorems above.

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


@@ -462,30 +475,60 @@ Let us finally remark that the vectorial negation satisfies the hypotheses of th
 
 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
+leading thus to a new PRNG that 
+\begin{color}{red}
+should improves the statistical properties of each
+generator taken alone. 
+Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input

 present.
 
 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)}
 $z\leftarrow{z\oplus{(z\ll13)}}$\;


@@ -493,37 +536,101 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\;
 $z\leftarrow{z\oplus{(z\ll5)}}$\;
 $y\leftarrow{z}$\;
 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 from changing twice a given
+bit between two outputs.
+This new generator is designed by the following process. 
+
+First of all, some chaotic iterations have to be done to generate a sequence 
+$\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ 
+of Boolean vectors, which are the successive states of the iterated system. 
+Some of these vectors will be randomly extracted and our pseudo-random bit 
+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 achieve chaotic iterations with
+the Boolean vectorial negation $f_0$ and some well-chosen strategies.
+Finally, some $x^n$ are selected
+by a sequence $m^n$ as the pseudo-random bit sequence of our generator.
+$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers.
+
+The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}.
+The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input
+PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}.
+This function is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$
+(the reader is referred to~\cite{bg10:ip} for more information).

 
 

+\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}

 
 

-
-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.
-
-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}
+\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$m\leftarrow{g(a)}$\;
+\STATE$k\leftarrow{m}$\;
+\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}
+\end{color}

 
 \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,\begin{color}{red} we now propose to choose a
+subset of components and to update them together, for speed improvements. Such a proposition leads\end{color}
+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}


@@ -533,17 +640,17 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N
 \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
 \end{array}
 \right.
 \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
 \end{array}
 \right.

-\label{equation Oplus}
+\label{equation Oplus0}

 \end{equation}
 where $\oplus$ is for the bitwise exclusive or between two integers. 
 \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.
 

-The single basic component presented in Eq.~\ref{equation Oplus} is of 
+The single basic component presented in Eq.~\ref{equation Oplus0} is of 

 ordinary use as a good elementary brick in various PRNGs. It corresponds
 to the following discrete dynamical system in chaotic iterations:
 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
 to the following discrete dynamical system in chaotic iterations:
 


@@ -560,14 +667,16 @@ 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 Oplus0}, 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


@@ -577,11 +686,11 @@ 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.


@@ -606,14 +715,13 @@ 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:


@@ -623,7 +731,7 @@ 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


@@ -640,7 +748,7 @@ 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 


@@ -650,17 +758,21 @@ 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
 $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
 


@@ -671,7 +783,7 @@ 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 $d$, as being the sum of two distances, will also be a distance.

  \begin{itemize}
 \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
 $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
  \begin{itemize}
 \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
 $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 


@@ -688,7 +800,7 @@ 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 


@@ -724,7 +836,7 @@ 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


@@ -739,14 +851,16 @@ 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
+$
+\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}
 


@@ -786,7 +900,7 @@ 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 the 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.


@@ -804,8 +918,10 @@ 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


@@ -813,51 +929,237 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \end{proof}
 
 
 \end{proof}
 
 

+\begin{color}{red}
+\section{Statistical Improvements Using Chaotic Iterations}
+
+\label{The generation of pseudo-random sequence}
+
+
+Let us now explain why we are reasonable grounds to believe that chaos 
+can improve statistical properties.
+We will show in this section that, when mixing defective PRNGs with
+chaotic iterations, the result presents better statistical properties
+(this section summarizes the work of~\cite{bfg12a:ip}).
+
+\subsection{Details of some Existing Generators}
+
+The list of defective PRNGs we will use 
+as inputs for the statistical tests to come is introduced here.
+
+Firstly, the simple linear congruency generator (LCGs) will be used. 
+It is defined by the following recurrence:
+\begin{equation}
+x^n = (ax^{n-1} + c)~mod~m
+\label{LCG}
+\end{equation}
+where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than 
+$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) 
+combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
+
+Secondly, the multiple recursive generators (MRGs) will be used too, which
+are based on a linear recurrence of order 
+$k$, modulo $m$~\cite{LEcuyerS07}:
+\begin{equation}
+x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m
+\label{MRG}
+\end{equation}
+Combination of two MRGs (referred as 2MRGs) is also used in these experimentations.
+
+Generators based on linear recurrences with carry will be regarded too.
+This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
+\begin{equation}
+\label{AWC}
+\begin{array}{l}
+x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
+c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
+the SWB generator, having the recurrence:
+\begin{equation}
+\label{SWB}
+\begin{array}{l}
+x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
+c^n=\left\{
+\begin{array}{l}
+1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
+0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
+and the SWC generator designed by R. Couture, which is based on the following recurrence:
+\begin{equation}
+\label{SWC}
+\begin{array}{l}
+x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
+c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
+
+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}

 
 

-\section{Efficient PRNG based on Chaotic Iterations}
+
+Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, 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}
+
+
+
+
+
+\subsection{Statistical tests}
+\label{Security analysis}
+
+Three batteries of tests are reputed and usually used
+to evaluate the statistical properties of newly designed pseudorandom
+number generators. These batteries are named DieHard~\cite{Marsaglia1996},
+the NIST suite~\cite{ANDREW2008}, and the most stringent one called
+TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
+
+
+
+\label{Results and discussion}
+\begin{table*}
+\renewcommand{\arraystretch}{1.3}
+\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
+\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
+\centering
+  \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
+    \hline\hline
+Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB  & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
+NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15}   & 14/15 & 14/15  & 14/15 & 14/15& 14/15& 14/15 \\ \hline
+DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
+\end{tabular}
+\end{table*}
+
+Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
+results on the two firsts batteries recalled above, indicating that all the PRNGs presented
+in the previous section
+cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot 
+fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
+iterations can solve this issue.
+%More precisely, to
+%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
+%\begin{enumerate}
+%  \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
+%  \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
+%  \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%\begin{equation}
+%\begin{array}{l}
+%\left\{
+%\begin{array}{l}
+%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
+%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
+%\end{equation}
+%$m$ is called the \emph{functional power}.
+%\end{enumerate}
+%
+The obtained results are reproduced in Table
+\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
+The scores written in boldface indicate that all the tests have been passed successfully, whereas an 
+asterisk ``*'' means that the considered passing rate has been improved.
+The improvements are obvious for both the ``Old CI'' and ``New CI'' generators.
+Concerning the ``Xor CI PRNG'', the speed improvement makes that statistical 
+results are not as good as for the two other versions of these CIPRNGs.
+
+
+\begin{table*}
+\renewcommand{\arraystretch}{1.3}
+\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
+\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
+\centering
+  \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
+    \hline
+Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG  & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
+Old CIPRNG\\ \hline \hline
+NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}   & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} *  & \textbf{18/18} *  & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
+New CIPRNG\\ \hline \hline
+NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}  & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
+Xor CIPRNG\\ \hline\hline
+NIST & 14/15*& \textbf{15/15} *   & \textbf{15/15}   & \textbf{15/15}   & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15}  \\ \hline
+DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
+\end{tabular}
+\end{table*}
+
+
+We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
+the statistical behavior of the Xor CI version by combining more than one 
+$\oplus$ operation. Results are summarized in~\ref{threshold}, showing
+that rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained 
+using chaotic iterations on defective generators.
+
+\begin{table*}
+\renewcommand{\arraystretch}{1.3}
+\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
+\label{threshold}
+\centering
+  \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
+    \hline
+Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3  & MRG2 \\ \hline\hline
+Threshold  value $m$& 19 & 7  & 2& 1 & 11& 9& 3& 4\\ \hline\hline
+\end{tabular}
+\end{table*}
+
+Next subsection gives a concrete implementation of this Xor CI PRNG, which will 
+new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does not
+raise ambiguity.
+\end{color}
+
+\subsection{Efficient Implementation of a PRNG based on Chaotic Iterations}

 \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, 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}

 
 

-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}
 
 
 
 \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();


@@ -872,8 +1174,7 @@ unsigned int CIPRNG() {
   return x;
 }
 \end{lstlisting}
   return x;
 }
 \end{lstlisting}

-
-
+\end{small}

 
 
 
 
 
 


@@ -887,7 +1188,7 @@ 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}.
 
 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}.
 

-So producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers
+Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers

 that  are provided by  3 64-bits  PRNGs.  This  version successfully  passes the
 stringent BigCrush battery of tests~\cite{LEcuyerS07}.
 
 that  are provided by  3 64-bits  PRNGs.  This  version successfully  passes the
 stringent BigCrush battery of tests~\cite{LEcuyerS07}.
 


@@ -905,7 +1206,7 @@ a   program    similar   to    the   one   presented    in  Listing
 do  so,  we  must   firstly  recall  that  in  the  CUDA~\cite{Nvid10}
 environment,    threads    have     a    local    identifier    called
 \texttt{ThreadIdx},  which   is  relative  to   the  block  containing
 do  so,  we  must   firstly  recall  that  in  the  CUDA~\cite{Nvid10}
 environment,    threads    have     a    local    identifier    called
 \texttt{ThreadIdx},  which   is  relative  to   the  block  containing

-them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU are
+them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU, are

 called {\it kernels}.
 
 
 called {\it kernels}.
 
 


@@ -913,10 +1214,10 @@ called {\it kernels}.
 
  
 It is possible to deduce from the CPU version a quite similar version adapted to GPU.
 
  
 It is possible to deduce from the CPU version a quite similar version adapted to GPU.

-The simple principle consists to make each thread of the GPU computing the CPU version of our PRNG.  
+The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG.  

 Of course,  the  three xor-like
 PRNGs  used in these computations must have different  parameters. 
 Of course,  the  three xor-like
 PRNGs  used in these computations must have different  parameters. 

-In a given thread, these lasts are
+In a given thread, these parameters are

 randomly picked from another PRNGs. 
 The  initialization stage is performed by  the CPU.
 To do it, the  ISAAC  PRNG~\cite{Jenkins96} is used to  set  all  the
 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


@@ -929,8 +1230,9 @@ 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\;}


@@ -943,14 +1245,16 @@ 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)},


@@ -961,14 +1265,14 @@ and  the pseudorandom  numbers generated by  our  PRNG,  is  equal to  $100,000\
 
 This generator is able to pass the whole BigCrush battery of tests, for all
 the versions that have been tested depending on their number of threads 
 
 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}


@@ -991,10 +1295,10 @@ been chosen. In practice, we  use the xor128 proposed in~\cite{Marsaglia2003} in
 which  unsigned longs  (64 bits)  have been  replaced by  unsigned  integers (32
 bits).
 
 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\;


@@ -1008,18 +1312,18 @@ array\_comb1, array\_comb2: Arrays containing combinations of size combination\_
   o2 = threadIdx-offset+array\_comb2[offset]\;
   \For{i=1 to n} {
     t=xor-like()\;
   o2 = threadIdx-offset+array\_comb2[offset]\;
   \For{i=1 to n} {
     t=xor-like()\;

-    t=t $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;
+    t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;

     shared\_mem[threadId]=t\;
     shared\_mem[threadId]=t\;

-    x = x $\wedge$ t\;
+    x = x\textasciicircum t\;

 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
   store internal variables in InternalVarXorLikeArray[threadId]\;
 }
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
   store internal variables in InternalVarXorLikeArray[threadId]\;
 }

-
-\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}
 
 \subsection{Theoretical Evaluation of the Improved Version}
 \end{algorithm}
 
 \subsection{Theoretical Evaluation of the Improved Version}


@@ -1033,7 +1337,7 @@ and two values previously obtained by two other threads).
 To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
 we must guarantee that this dynamical system iterates on the space 
 $\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
 To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
 we must guarantee that this dynamical system iterates on the space 
 $\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.

-The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+The left term $x$ obviously belongs to $\mathds{B}^ \mathsf{N}$.

 To prevent from any flaws of chaotic properties, we must check that the right 
 term (the last $t$), corresponding to the strategies,  can possibly be equal to any
 integer of $\llbracket 1, \mathsf{N} \rrbracket$. 
 To prevent from any flaws of chaotic properties, we must check that the right 
 term (the last $t$), corresponding to the strategies,  can possibly be equal to any
 integer of $\llbracket 1, \mathsf{N} \rrbracket$. 


@@ -1070,7 +1374,7 @@ into the GPU memory has been removed. This step is time consuming and slows down
 generation.  Moreover this   storage  is  completely
 useless, in case of applications that consume the pseudorandom
 numbers  directly   after generation. We can see  that when the number of  threads is greater
 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
 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


@@ -1081,7 +1385,7 @@ As a  comparison,   Listing~\ref{algo:seqCIPRNG}  leads   to the  generation of
 
 \begin{figure}[htbp]
 \begin{center}
 
 \begin{figure}[htbp]
 \begin{center}

-  \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf}
+  \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf}

 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
 \label{fig:time_xorlike_gpu}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
 \label{fig:time_xorlike_gpu}


@@ -1091,17 +1395,16 @@ 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[width=\columnwidth]{curve_time_bbs_gpu.pdf}

 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
 \label{fig:time_bbs_gpu}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
 \label{fig:time_bbs_gpu}


@@ -1109,7 +1412,7 @@ reduction.
 
 All  these  experiments allow  us  to conclude  that  it  is possible  to
 generate a very large quantity of pseudorandom  numbers statistically perfect with the  xor-like version.
 
 All  these  experiments allow  us  to conclude  that  it  is possible  to
 generate a very large quantity of pseudorandom  numbers statistically perfect with the  xor-like version.

-In a certain extend, it is the case too with the secure BBS-based version, the speed deflation being
+To a certain extend, it is also the case with the secure BBS-based version, the speed deflation being

 explained by the fact that the former  version has ``only''
 chaotic properties and statistical perfection, whereas the latter is also cryptographically secure,
 as it is shown in the next sections.
 explained by the fact that the former  version has ``only''
 chaotic properties and statistical perfection, whereas the latter is also cryptographically secure,
 as it is shown in the next sections.


@@ -1129,17 +1432,17 @@ In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any
 denoted by $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any

-seed $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size
-$\ell_G(k)$ with $\ell_G(k)>k$.
+seed $s$ of length $m$, $G(s)$ (the output of $G$ on the input $s$) has size
+$\ell_G(m)$ with $\ell_G(m)>m$.

 The notion of {\it secure} PRNGs can now be defined as follows. 
 
 \begin{definition}
 A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
 algorithm $D$, for any positive polynomial $p$, and for all sufficiently
 The notion of {\it secure} PRNGs can now be defined as follows. 
 
 \begin{definition}
 A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
 algorithm $D$, for any positive polynomial $p$, and for all sufficiently

-large $k$'s,
-$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$
+large $m$'s,
+$$| \mathrm{Pr}[D(G(U_m))=1]-Pr[D(U_{\ell_G(m)})=1]|< \frac{1}{p(m)},$$

 where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the
 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}
 


@@ -1148,7 +1451,7 @@ distinguish a perfect uniform random generator from $G$ with a non
 negligible probability. The interested reader is referred
 to~\cite[chapter~3]{Goldreich} for more information. Note that it is
 quite easily possible to change the function $\ell$ into any polynomial
 negligible probability. The interested reader is referred
 to~\cite[chapter~3]{Goldreich} for more information. Note that it is
 quite easily possible to change the function $\ell$ into any polynomial

-function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}.
+function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.

 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,
 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,


@@ -1159,7 +1462,7 @@ 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.
 


@@ -1203,8 +1506,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. 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 $x$,


@@ -1213,7 +1518,7 @@ 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. 


@@ -1223,11 +1528,11 @@ It follows that
 \mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1].
 \end{equation}
  From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that
 \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}
 
 


@@ -1240,7 +1545,7 @@ 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:
+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. 
 $$x_{n+1}=x_n^2~ mod~ M$$  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. 


@@ -1248,37 +1553,37 @@ 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
   
 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
+required to use only modulus  on 32-bits integer numbers. Consequently

 $x_n^2$ need  to be lesser than $2^{32}$,  and thus the number $M$ must be
 lesser than $2^{16}$.  So in practice we can choose prime numbers around
 $x_n^2$ need  to be lesser than $2^{32}$,  and thus the number $M$ must be
 lesser 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
+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

-$log_2(log_2(M))$). In other words, to generate a  32 bits number, we need to use
+$log_2(log_2(M))$). 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
 8 times  the BBS  algorithm with possibly different  combinations of  $M$. This

-approach is  not sufficient to be able to pass  all the TestU01,
+approach is  not sufficient to be able to pass  all the tests of TestU01,

 as small values of  $M$ for the BBS  lead to
 as small values of  $M$ for the BBS  lead to

-  small periods. So, in  order to add randomness  we proceed with
+  small periods. So, in  order to add randomness  we have proceeded with

 the followings  modifications. 
 \begin{itemize}
 \item
 Firstly, we  define 16 arrangement arrays  instead of 2  (as described in
 Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of
 the followings  modifications. 
 \begin{itemize}
 \item
 Firstly, we  define 16 arrangement arrays  instead of 2  (as described in
 Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of

-the  PRNG kernels. In  practice, the  selection of   combinations
+the  PRNG kernels. In  practice, the  selection of   combination

 arrays to be used is different for all the threads. It is determined
 by using  the three last bits  of two internal variables  used by BBS.
 %This approach  adds more randomness.   
 In Algorithm~\ref{algo:bbs_gpu},
 character  \& is for the  bitwise AND. Thus using  \&7 with  a number
 arrays to be used is different for all the threads. It is determined
 by using  the three last bits  of two internal variables  used by BBS.
 %This approach  adds more randomness.   
 In Algorithm~\ref{algo:bbs_gpu},
 character  \& is for the  bitwise AND. Thus using  \&7 with  a number

-gives the last 3 bits, providing so a number between 0 and 7.
+gives the last 3 bits, thus providing a number between 0 and 7.

 \item
 Secondly, after the  generation of the 8 BBS numbers  for each thread, we
 \item
 Secondly, after the  generation of the 8 BBS numbers  for each thread, we

-have a 32 bits number whose period is possibly quite small. So
+have a 32-bits number whose period is possibly quite small. So

 to add randomness,  we generate 4 more BBS numbers   to
 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
+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
 described  in Algorithm~\ref{algo:bbs_gpu}.  In  practice, the last 2 bits
 of the first new BBS number are  used to make a left shift of at most

-3 bits. The  last 3 bits of the  second new BBS number are  add to the
+3 bits. The  last 3 bits of the  second new BBS number are  added to the

 strategy whatever the value of the first left shift. The third and the
 fourth new BBS  numbers are used similarly to apply  a new left shift
 and add 3 new bits.
 strategy whatever the value of the first left shift. The third and the
 fourth new BBS  numbers are used similarly to apply  a new left shift
 and add 3 new bits.


@@ -1291,12 +1596,12 @@ variable for BBS number 8 is stored in place 1.
 \end{itemize}
 
 \begin{algorithm}
 \end{itemize}
 
 \begin{algorithm}

-
+\begin{small}

 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 in global memory\;
 NumThreads: Number of threads\;
 array\_comb: 2D Arrays containing 16 combinations (in first dimension)  of size combination\_size (in second dimension)\;
 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 in global memory\;
 NumThreads: Number of threads\;
 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}


@@ -1307,57 +1612,92 @@ array\_comb: 2D Arrays containing 16 combinations (in first dimension)  of size
   o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
   o2 = threadIdx-offset+array\_comb[8+bbs2\&7][offset]\;
   \For{i=1 to n} {
   o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
   o2 = threadIdx-offset+array\_comb[8+bbs2\&7][offset]\;
   \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 \^{ } shmem[o1] \^{ }    shmem[o2]\;
+    \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[threadId]=t\;
     shared\_mem[threadId]=t\;

-    x = x \^{ }    t\;
+    x = x\textasciicircum   t\;

 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
   store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
 }
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
   store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
 }

-
+\end{small}

 \caption{main kernel for the BBS based PRNG GPU}
 \label{algo:bbs_gpu}
 \end{algorithm}
 
 \caption{main kernel for the BBS based PRNG GPU}
 \label{algo:bbs_gpu}
 \end{algorithm}
 

-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 to initialize $t$ 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 and we put \emph{exactly} the 3 last bits from a BBS into 
-the 3 last bits of $t$, leading possibly to a loss of a few 
-bits of $t$. 
-
-It should  be noticed that this generator has another time the form $x^{n+1} = x^n \oplus S^n$,
+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
 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
+secure.

 
 
 
 
 
 

+\begin{color}{red}
+\subsection{Practical Security Evaluation}
+
+Suppose now that the PRNG will work during 
+$M=100$ time units, and that during this period,
+an attacker can realize $10^{12}$ clock cycles.
+We thus wonder whether, during the PRNG's 
+lifetime, the attacker can distinguish this 
+sequence from truly random one, with a probability
+greater than $\varepsilon = 0.2$.
+We consider that $N$ has 900 bits.
+
+The random process is the BBS generator, which
+is cryptographically secure. More precisely, it
+is $(T,\varepsilon)-$secure: no 
+$(T,\varepsilon)-$distinguishing attack can be
+successfully realized on this PRNG, if~\cite{Fischlin}
+$$
+T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
+$$
+where $M$ is the length of the output ($M=100$ in
+our example), and $L(N)$ is equal to
+$$
+2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)
+$$
+is the number of clock cycles to factor a $N-$bit
+integer.
+
+A direct numerical application shows that this attacker 
+cannot achieve its $(10^{12},0.2)$ distinguishing
+attack in that context.
+
+\end{color}
+

 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
 \label{Blum-Goldwasser}
 We finish this research work by giving some thoughts about the use of
 \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


@@ -1401,7 +1741,7 @@ When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$
 \item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
 \item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$.
 \item She recomputes the bit-vector $b$ by using BBS and $x_0$.
 \item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
 \item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$.
 \item She recomputes the bit-vector $b$ by using BBS and $x_0$.

-\item Alice computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.
+\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.

 \end{enumerate}
 
 
 \end{enumerate}
 
 


@@ -1414,14 +1754,16 @@ Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too
 her new public key will be $(S^0, N)$.
 
 To encrypt his message, Bob will compute
 her new public key will be $(S^0, N)$.
 
 To encrypt his message, Bob will compute

-\begin{equation}
-c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)
-\end{equation}
+%%RAPH : ici, j'ai mis un simple $
+%\begin{equation}
+$c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.$
+$ \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)$
+%%\end{equation}

 instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. 
 
 The same decryption stage as in Blum-Goldwasser leads to the sequence 
 $\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.
 instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. 
 
 The same decryption stage as in Blum-Goldwasser leads to the sequence 
 $\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.

-Thus, with a simple use of $S^0$, Alice can obtained the plaintext.
+Thus, with a simple use of $S^0$, Alice can obtain the plaintext.

 By doing so, the proposed generator is used in place of BBS, leading to
 the inheritance of all the properties presented in this paper.
 
 By doing so, the proposed generator is used in place of BBS, leading to
 the inheritance of all the properties presented in this paper.
 


@@ -1432,16 +1774,16 @@ 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
 has been generalized to improve its speed. It has been proven to be
 chaotic according to Devaney.
 Efficient implementations on  GPU using xor-like  PRNGs as input generators

-shown that a very large quantity of pseudorandom numbers can be generated per second (about
+have shown that a very large quantity of pseudorandom numbers can be generated per second (about

 20Gsamples/s), and that these proposed PRNGs succeed to pass the hardest battery in TestU01,
 namely the BigCrush.
 Furthermore, we have shown that when the inputted generator is cryptographically
 secure, then it is the case too for the PRNG we propose, thus leading to
 the possibility to develop fast and secure PRNGs using the GPU architecture.
 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 to develop fast and secure PRNGs using the GPU architecture.

-Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the 
-proposed method, has been finally proposed.
+\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it 
+behaves chaotically, has finally been proposed. \end{color}

 
 

-In future  work we plan to extend these researches, building a parallel PRNG for  clusters or
+In future  work we plan to extend this research, building a parallel PRNG for  clusters or

 grid computing. Topological properties of the various proposed generators will be investigated,
 and the use of other categories of PRNGs as input will be studied too. The improvement
 of Blum-Goldwasser will be deepened. Finally, we
 grid computing. Topological properties of the various proposed generators will be investigated,
 and the use of other categories of PRNGs as input will be studied too. The improvement
 of Blum-Goldwasser will be deepened. Finally, we