Ajout e la preuve de pch

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 081ec9ca9e4a508e145b295fc1bdba8b1f48b174..d2e5e0547d736054ce547ef49dd39e7b5f0c1a3c 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -34,66 +34,176 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 

-\title{Efficient generation of pseudo random numbers based on chaotic iterations on GPU}
+\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
+on GPU}

 \begin{document}
 
 \begin{document}
 

-\author{Jacques M. Bahi, Rapha\"{e}l Couturier, and Christophe Guyeux\thanks{Authors in alphabetic order}}
+\author{Jacques M. Bahi, Rapha\"{e}l Couturier, and Christophe
+Guyeux, Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}

 
 \maketitle
 
 \begin{abstract}
 
 \maketitle
 
 \begin{abstract}

-This is the abstract
+In this paper we present a new pseudo-random numbers generator (PRNG) on
+graphics processing units  (GPU). This PRNG is based  on chaotic iterations.  it
+is proven  to be chaotic  in the Devanay's  formulation. We propose  an efficient
+implementation  for  GPU which  succeeds  to  the  {\it BigCrush},  the  hardest
+batteries of test of TestU01.  Experimentations show that this PRNG can generate
+about 20 billions of random numbers  per second on Tesla C1060 and NVidia GTX280
+cards.
+
+

 \end{abstract}
 
 \section{Introduction}
 
 \end{abstract}
 
 \section{Introduction}
 

-Interet des itérations chaotiques pour générer des nombre alea\\
-Interet de générer des nombres alea sur GPU
-\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
-...
-
+Random  numbers are  used in  many scientific  applications and  simulations. On
+finite  state machines,  as computers,  it is  not possible  to  generate random
+numbers but only pseudo-random numbers. In practice, a good pseudo-random numbers
+generator (PRNG) needs  to verify some features to be used  by scientists. It is
+important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
+generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
+statistical properties. Finally, from our point a view, it is essential to prove
+that  a PRNG  is  chaotic.  Concerning  the  statistical tests,  TestU01 is  the
+best-known public-domain statistical testing package.   So we use it for all our
+PRNGs, especially the {\it BigCrush}  which provides the largest serie of tests.
+Concerning  the  chaotic properties,  Devaney~\cite{Devaney}  proposed a  common
+mathematical formulation of chaotic dynamical systems.
+
+In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic
+PRNG  based on  chaotic iterations. We  have proven  that these  PRNGs are
+chaotic in the Devaney's sense.  In this paper we propose a faster version which
+is also proven to be chaotic.
+
+Although graphics  processing units (GPU)  was initially designed  to accelerate
+the manipulation of  images, they are nowadays commonly  used in many scientific
+applications. Therefore,  it is important  to be able to  generate pseudo-random
+numbers inside a GPU when a scientific application runs in a GPU. That is why we
+also provide  an efficient  PRNG for  GPU respecting based  on IC.  Such devices
+allows us to generated almost 20 billions of random numbers per second.
+
+In order  to establish  that our  PRNGs are chaotic  according to  the Devaney's
+formulation, we  extend what we  have proposed in~\cite{guyeux10}.
+
+The rest of this paper  is organised as follows. In Section~\ref{section:related
+  works} we  review some GPU implementions  of PRNG.  Section~\ref{section:BASIC
+  RECALLS} gives some basic recalls  on Devanay's formation of chaos and chaotic
+iterations. In  Section~\ref{sec:pseudo-random} the proof of chaos  of our PRNGs
+is   studied.    Section~\ref{sec:efficient    prng}   presents   an   efficient
+implementation of  our chaotic PRNG  on a CPU.   Section~\ref{sec:efficient prng
+  gpu}   describes   the  GPU   implementation   of   our   chaotic  PRNG.    In
+Section~\ref{sec:experiments}     some    experimentations     are    presented.
+ Finally, we give a conclusion and some perspectives.
+
+
+
+
+\section{Related works on GPU based PRNGs}
+\label{section:related works}
+In the litterature many authors have work on defining GPU based PRNGs. We do not
+want to be exhaustive and we just give the most significant works from our point
+of view. When authors mention the  number of random numbers generated per second
+we mention  it. We  consider that  a million numbers  per second  corresponds to
+1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
+
+In \cite{Pang:2008:cec},  the authors define  a PRNG based on  cellular automata
+which  does   not  require  high  precision  integer   arithmetics  nor  bitwise
+operations. There is no mention of statistical tests nor proof that this PRNG is
+chaotic.  Concerning   the  speed  of   generation,  they  can   generate  about
+3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
+
+In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
+based on  Lagged Fibonacci, Hybrid  Taus 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
+CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
+no mention that their PRNGs have chaos mathematical properties.
+
+
+Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
+PRNGs on  diferrent computing architectures: CPU,  field-programmable gate array
+(FPGA), GPU and massively parallel  processor. This study is interesting because
+it  shows the  performance  of the  same  PRNGs on  different architeture.   For
+example,  the FPGA  is globally  the  fastest architecture  and it  is also  the
+efficient one because it provides the fastest number of generated random numbers
+per joule. Concerning the GPU,  authors can generate betweend 11 and 16GSample/s
+with a GTX 280  GPU. The drawback of this work is  that those PRNGs only succeed
+the {\it Crush} test which is easier than the {\it Big Crush} test.
+
+Cuda  has developped  a  library for  the  generation of  random numbers  called
+Curand~\cite{curand11}.        Several       PRNGs        are       implemented:
+Xorwow~\cite{Marsaglia2003} and  some variants of Sobol. Some  tests report that
+the  fastest version provides  15GSample/s on  the new  Fermi C2050  card. Their
+PRNGs fail to succeed the whole tests of TestU01 on only one test.
+\newline
+\newline
+To the best of our knowledge no GPU implementation have been proven to have chaotic properties.

 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}

-This section is devoted to basic definitions and terminologies in the fields of topological chaos and chaotic iterations.
-\subsection{Devaney's chaotic dynamical systems}
+This section is devoted to basic definitions and terminologies in the fields of
+topological chaos and chaotic iterations.
+\subsection{Devaney's Chaotic Dynamical Systems}

 
 

-In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$ denotes the $k^{th}$ composition of a function $f$. Finally, the following notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
+In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
+denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
+is for the $k^{th}$ composition of a function $f$. Finally, the following
+notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.

 
 
 
 

-Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : \mathcal{X} \rightarrow \mathcal{X}$.
+Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f :
+\mathcal{X} \rightarrow \mathcal{X}$.

 
 \begin{definition}
 
 \begin{definition}

-$f$ 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 $f^k(U) \cap V \neq \varnothing$.
+$f$ 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}
 
 \begin{definition}
 \end{definition}
 
 \begin{definition}

-An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$ if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$
+An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$
+if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$

 \end{definition}
 
 \begin{definition}
 \end{definition}
 
 \begin{definition}

-$f$ is said to be \emph{regular} on $(\mathcal{X}, \tau)$ if the set of periodic points for $f$ is dense in $\mathcal{X}$: for any point $x$ in $\mathcal{X}$, any neighborhood of $x$ contains at least one periodic point (without necessarily the same period).
+$f$ is said to be \emph{regular} on $(\mathcal{X}, \tau)$ if the set of periodic
+points for $f$ is dense in $\mathcal{X}$: for any point $x$ in $\mathcal{X}$,
+any neighborhood of $x$ contains at least one periodic point (without
+necessarily the same period).

 \end{definition}
 
 
 \end{definition}
 
 

-\begin{definition}
-$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and topologically transitive.
+\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
+topologically transitive.

 \end{definition}
 
 \end{definition}
 

-The chaos property is strongly linked to the notion of ``sensitivity'', defined on a metric space $(\mathcal{X},d)$ by:
+The chaos property is strongly linked to the notion of ``sensitivity'', defined
+on a metric space $(\mathcal{X},d)$ by:

 
 \begin{definition}
 \label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
 
 \begin{definition}
 \label{sensitivity} $f$ 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 $f$.
 \end{definition}
 
 
 $\delta$ 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 chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has the property of sensitive dependence on initial conditions (this property was formerly an element of the definition of chaos). To sum up, quoting Devaney in~\cite{Devaney}, a chaotic dynamical system ``is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or simplified into two subsystems which do not interact because of topological transitivity. And in the midst of this random behavior, we nevertheless have an element of regularity''. Fundamentally different behaviors are consequently possible and occur in an unpredictable way.
+Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is
+chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has the property of
+sensitive dependence on initial conditions (this property was formerly an
+element of the definition of chaos). To sum up, quoting Devaney
+in~\cite{Devaney}, a chaotic dynamical system ``is unpredictable because of the
+sensitive dependence on initial conditions. It cannot be broken down or
+simplified into two subsystems which do not interact because of topological
+transitivity. And in the midst of this random behavior, we nevertheless have an
+element of regularity''. Fundamentally different behaviors are consequently
+possible and occur in an unpredictable way.

 
 
 
 
 
 

-\subsection{Chaotic iterations}
+\subsection{Chaotic Iterations}

 \label{sec:chaotic iterations}
 
 
 \label{sec:chaotic iterations}
 
 


@@ -103,23 +213,23 @@ Boolean  \emph{state}. Having $\mathsf{N}$ Boolean values for these
  cells  leads to the definition of a particular \emph{state  of the
 system}. A sequence which  elements belong to $\llbracket 1;\mathsf{N}
 \rrbracket $ is called a \emph{strategy}. The set of all strategies is
  cells  leads to the definition of a particular \emph{state  of the
 system}. A sequence which  elements belong to $\llbracket 1;\mathsf{N}
 \rrbracket $ is called a \emph{strategy}. The set of all strategies is

-denoted by $\mathbb{S}.$
+denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$

 
 \begin{definition}
 \label{Def:chaotic iterations}
 The      set       $\mathds{B}$      denoting      $\{0,1\}$,      let
 $f:\mathds{B}^{\mathsf{N}}\longrightarrow  \mathds{B}^{\mathsf{N}}$ be
 
 \begin{definition}
 \label{Def:chaotic iterations}
 The      set       $\mathds{B}$      denoting      $\{0,1\}$,      let
 $f:\mathds{B}^{\mathsf{N}}\longrightarrow  \mathds{B}^{\mathsf{N}}$ be

-a  function  and  $S\in  \mathbb{S}$  be  a  strategy.  The  so-called
+a  function  and  $S\in  \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$  be  a  ``strategy''.  The  so-called

 \emph{chaotic      iterations}     are     defined      by     $x^0\in
 \mathds{B}^{\mathsf{N}}$ and
 \emph{chaotic      iterations}     are     defined      by     $x^0\in
 \mathds{B}^{\mathsf{N}}$ and

-$$
+\begin{equation}

 \forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
 \llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
 \begin{array}{ll}
   x_i^{n-1} &  \text{ if  }S^n\neq i \\
   \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.
 \end{array}\right.
 \forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
 \llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
 \begin{array}{ll}
   x_i^{n-1} &  \text{ if  }S^n\neq i \\
   \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.
 \end{array}\right.

-$$
+\end{equation}

 \end{definition}
 
 In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
 \end{definition}
 
 In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is


@@ -129,49 +239,59 @@ $\left(f(x^{n-1})\right)_{S^{n}}$      can     be      replaced     by
 $\left(f(x^{k})\right)_{S^{n}}$, where  $k<n$, describing for example,
 delays  transmission~\cite{Robert1986,guyeux10}.  Finally,  let us  remark that
 the term  ``chaotic'', in  the name of  these iterations,  has \emph{a
 $\left(f(x^{k})\right)_{S^{n}}$, where  $k<n$, describing for example,
 delays  transmission~\cite{Robert1986,guyeux10}.  Finally,  let us  remark that
 the term  ``chaotic'', in  the name of  these iterations,  has \emph{a

-priori} no link with the mathematical theory of chaos, recalled above.
+priori} no link with the mathematical theory of chaos, presented above.

 
 
 
 

-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 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}, $\delta (x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
-\begin{equation*}
+Let $\delta $ be the \emph{discrete Boolean metric}, $\delta
+(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
+\begin{equation}

 \begin{array}{lrll}
 F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
 \longrightarrow & \mathds{B}^{\mathsf{N}} \\
 & (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
 (k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
 \end{array}%
 \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},%
 \end{array}%

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

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

-\begin{equation*}
+\begin{equation}

 \mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
 \mathds{B}^\mathsf{N},
 \mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
 \mathds{B}^\mathsf{N},

-\end{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}
 \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}
 \end{equation}

-\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma (S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in \mathds{N}}\in \mathbb{S}$ and $i$ is the \emph{initial function}  $i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket 1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in (\ref{sec:chaotic iterations}) can be described by the following iterations:
-\begin{equation*}
+\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
+(S^{n})_{n\in \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow (S^{n+1})_{n\in
+\mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ and $i$ is the \emph{initial function} 
+$i:(S^{n})_{n\in \mathds{N}} \in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow S^{0}\in \llbracket
+1;\mathsf{N}\rrbracket$. Then the chaotic iterations proposed in
+Definition \ref{Def:chaotic iterations} can be described by the following iterations:
+\begin{equation}

 \left\{
 \begin{array}{l}
 X^0 \in \mathcal{X} \\
 X^{k+1}=G_{f}(X^k).%
 \end{array}%
 \right.
 \left\{
 \begin{array}{l}
 X^0 \in \mathcal{X} \\
 X^{k+1}=G_{f}(X^k).%
 \end{array}%
 \right.

-\end{equation*}%
-
-With this formulation, a shift function appears as a component of chaotic iterations. The shift function is a famous example of a chaotic map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as chaotic. 
-
-To study this claim, a new distance between two points $X = (S,E), Y = (\check{S},\check{E})\in
+\end{equation}%
+
+With this formulation, a shift function appears as a component of chaotic
+iterations. The shift function is a famous example of a chaotic
+map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as
+chaotic. 
+To study this claim, a new distance between two points $X = (S,E), Y =
+(\check{S},\check{E})\in

 \mathcal{X}$ has been introduced in \cite{guyeux10} as follows:
 \mathcal{X}$ has been introduced in \cite{guyeux10} as follows:

-\begin{equation*}
+\begin{equation}

 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),

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

 \noindent where
 \noindent where

-\begin{equation*}
+\begin{equation}

 \left\{
 \begin{array}{lll}
 \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
 \left\{
 \begin{array}{lll}
 \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%


@@ -180,62 +300,88 @@ d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
 \sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
 \end{array}%
 \right.
 \sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
 \end{array}%
 \right.

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

 
 
 This new distance has been introduced to satisfy the following requirements.
 \begin{itemize}
 
 
 This new distance has been introduced to satisfy the following requirements.
 \begin{itemize}

-\item When the number of different cells between two systems is increasing, then their distance should increase too.
-\item In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a while. Indeed, the two dynamical systems start with the same initial condition, use the same update function, and as strategies are the same for a while, then components that are updated are the same too.
+\item When the number of different cells between two systems is increasing, then
+their distance should increase too.
+\item In addition, if two systems present the same cells and their respective
+strategies start with the same terms, then the distance between these two points
+must be small because the evolution of the two systems will be the same for a
+while. Indeed, the two dynamical systems start with the same initial condition,
+use the same update function, and as strategies are the same for a while, then
+components that are updated are the same too.

 \end{itemize}
 \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 $E, \check{E}$ differ in $n$ cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a measure of the differences between strategies $S$ and $\check{S}$. More precisely, this floating part is less than $10^{-k}$ if and only if the first $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the $k^{th}$ terms of the two strategies are different.
+The distance presented above follows these recommendations. Indeed, if the floor
+value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$
+differ in $n$ cells ($d_e$ is indeed the Hamming distance). In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a
+measure of the differences between strategies $S$ and $\check{S}$. More
+precisely, this floating part is less than $10^{-k}$ if and only if the first
+$k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is
+nonzero, then the $k^{th}$ terms of the two strategies are different.
+The impact of this choice for a distance will be investigate at the end of the document.

 
 Finally, it has been established in \cite{guyeux10} that,
 
 \begin{proposition}
 
 Finally, it has been established in \cite{guyeux10} that,
 
 \begin{proposition}

-Let $f$ be a map from $\mathds{B}^n$ to itself. Then $G_{f}$ is continuous in the metric space $(\mathcal{X},d)$.
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. Then $G_{f}$ is continuous in
+the metric space $(\mathcal{X},d)$.

 \end{proposition}
 
 \end{proposition}
 

-The chaotic property of $G_f$ has been firstly established for the vectorial Boolean negation \cite{guyeux10}. To obtain a characterization, we have secondly introduced the notion of asynchronous iteration graph recalled bellow.
+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
+introduced the notion of asynchronous iteration graph recalled bellow.

 
 

-Let $f$ be a map from $\mathds{B}^n$ to itself. The
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The

 {\emph{asynchronous iteration graph}} associated with $f$ is the
 directed graph $\Gamma(f)$ defined by: the set of vertices is
 {\emph{asynchronous iteration graph}} associated with $f$ is the
 directed graph $\Gamma(f)$ defined by: the set of vertices is

-$\mathds{B}^n$; for all $x\in\mathds{B}^n$ and $i\in \llbracket1;n\rrbracket$,
+$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and 
+$i\in \llbracket1;\mathsf{N}\rrbracket$,

 the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. 
 The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
 strategy $s$ such that the parallel iteration of $G_f$ from the
 initial point $(s,x)$ reaches the point $x'$.
 
 the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. 
 The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
 strategy $s$ such that the parallel iteration of $G_f$ from the
 initial point $(s,x)$ reaches the point $x'$.
 

-We have finally proven in \cite{FCT11} that,
+We have finally proven in \cite{bcgr11:ip} that,

 
 
 \begin{theorem}
 \label{Th:Caractérisation   des   IC   chaotiques}  
 
 
 \begin{theorem}
 \label{Th:Caractérisation   des   IC   chaotiques}  

-Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic  (according to  Devaney) 
+Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic  (according to  Devaney) 

 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 

-This result of chaos has lead us to study the possibility to build a pseudo-random number generator (PRNG) based on the chaotic iterations. 
-As $G_f$, defined on the domain   $\llbracket 1 ;  n \rrbracket^{\mathds{N}}  \times \mathds{B}^n$, is build from Boolean networks $f : \mathds{B}^n \rightarrow \mathds{B}^n$, we can preserve the theoretical properties on $G_f$ during implementations (due to the discrete nature of $f$). It is as if $\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ;  n \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
+This result of chaos has lead us to study the possibility to build a
+pseudo-random 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 $f : \mathds{B}^\mathsf{N}
+\rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
+during implementations (due to the discrete nature of $f$). It is as if
+$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
+\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).

 
 \section{Application to Pseudo-Randomness}
 
 \section{Application to Pseudo-Randomness}

+\label{sec:pseudo-random}
+\subsection{A First Pseudo-Random Number Generator}

 
 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 
 
 We have proposed in~\cite{bgw09:ip} a new family of generators that receives 
 two PRNGs as inputs. These two generators are mixed with chaotic iterations, 
 leading thus to a new PRNG that improves the statistical properties of each
 generator taken alone. Furthermore, our generator 

-possesses various chaos properties
-that none of the generators used as input present.
+possesses various chaos properties that none of the generators used as input
+present.

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

-\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)}
+\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
+($n$ bits)}

 \KwOut{a configuration $x$ ($n$ bits)}
 $x\leftarrow x^0$\;
 \KwOut{a configuration $x$ ($n$ bits)}
 $x\leftarrow x^0$\;

-$k\leftarrow b + \textit{XORshift}(b+1)$\;
-\For{$i=0,\dots,k-1$}
+$k\leftarrow b + \textit{XORshift}(b)$\;
+\For{$i=0,\dots,k$}

 {
 $s\leftarrow{\textit{XORshift}(n)}$\;
 $x\leftarrow{F_f(s,x)}$\;
 {
 $s\leftarrow{\textit{XORshift}(n)}$\;
 $x\leftarrow{F_f(s,x)}$\;


@@ -247,7 +393,6 @@ return $x$\;
 \end{algorithm}
 
 \begin{algorithm}[h!]
 \end{algorithm}
 
 \begin{algorithm}[h!]

-%\SetAlgoLined                        %%RAPH: cette ligne provoque une erreur chez moi

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


@@ -266,15 +411,20 @@ return $y$\;
 
 This generator is synthesized in Algorithm~\ref{CI Algorithm}.
 It takes as input: a function $f$;
 
 This generator is synthesized in Algorithm~\ref{CI Algorithm}.
 It takes as input: a function $f$;

-an integer $b$, ensuring that the number of executed iterations is at least $b$ and at most $2b+1$; and an initial configuration $x^0$.
+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
 It returns the new generated configuration $x$.  Internally, it embeds two

-\textit{XORshift}$(k)$ PRNGs \cite{Marsaglia2003} that returns integers uniformly distributed
+\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that returns integers
+uniformly distributed

 into $\llbracket 1 ; k \rrbracket$.
 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.
+\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia,
+which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
+with a bit shifted version of it. This PRNG, which has a period of
+$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
+in our PRNG to compute the strategy length and the strategy elements.

 
 
 
 

-We have proven in \cite{FCT11} that,
-
+We have proven in \cite{bcgr11:ip} that,

 \begin{theorem}
   Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
   iteration graph, $\check{M}$ its adjacency
 \begin{theorem}
   Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
   iteration graph, $\check{M}$ its adjacency


@@ -285,504 +435,1214 @@ We have proven in \cite{FCT11} that,
   if and only if $M$ is a double stochastic matrix.
 \end{theorem} 
 
   if and only if $M$ is a double stochastic matrix.
 \end{theorem} 
 

+This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.

 
 

+\subsection{Improving the Speed of the Former Generator}

 
 

-\alert{Mettre encore un peu de blabla sur le PRNG, puis enchaîner en disant que, ok, on peut préserver le chaos quand on passe sur machine, mais que le chaos dont il s'agit a été prouvé pour une distance bizarroïde sur un espace non moins hémoroïde, d'où ce qui suit}
-
+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,
+this algorithm can be rewritten as follows:

 
 

+\begin{equation}
+\left\{
+\begin{array}{l}
+x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
+\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
+\end{array}
+\right.
+\label{equation Oplus}
+\end{equation}
+where $\oplus$ is for the bitwise exclusive or between two integers. 
+This rewritten can be understood as follows. The $n-$th term $S^n$ of the
+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 
+ordinary use as a good elementary brick in various PRNGs. It corresponds
+to the following discrete dynamical system in chaotic iterations:

 
 

-\section{The relativity of disorder}
-\label{sec:de la relativité du désordre}
-
-\subsection{Impact of the topology's finenesse}
-
-Let us firstly introduce the following notations.
+\begin{equation}
+\forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
+\llbracket1;\mathsf{N}\rrbracket ,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.
+\end{array}\right.
+\label{eq:generalIC}
+\end{equation}
+where $f$ is the vectorial negation and $\forall n \in \mathds{N}$, 
+$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
+$k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary
+decomposition of $S^n$ is 1. Such chaotic iterations are more general
+than the ones presented in Definition \ref{Def:chaotic iterations} for 
+the fact that, instead of updating only one term at each iteration,
+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
+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
+use of more general chaotic iterations to generate pseudo-random numbers 
+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 
+the general form:

 
 

-\begin{notation}
-$\mathcal{X}_\tau$ will denote the topological space $\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply $\mathcal{V} (x)$, if there is no ambiguity).
-\end{notation}
+\begin{equation}
+\forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
+\llbracket1;\mathsf{N}\rrbracket ,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.
+\end{array}\right.
+\label{general CIs}
+\end{equation}

 
 

+In other words, at the $n^{th}$ iteration, only the cells whose id is
+contained into the set $S^{n}$ are iterated.

 
 

+Let us now rewrite these general chaotic iterations as usual discrete dynamical
+system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
+is required in order to study the topological behavior of the system.

 
 

-\begin{theorem}
-\label{Th:chaos et finesse}
-Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t. $\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous both for $\tau$ and $\tau'$.
+Let us introduce the following function:
+\begin{equation}
+\begin{array}{cccc}
+ \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
+         & (i,X) & \longmapsto  & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X,  \end{array}\right.
+\end{array} 
+\end{equation}
+where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.

 
 

-If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then $(\mathcal{X}_\tau,f)$ is chaotic too.
-\end{theorem}
+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},%
+\end{array}%
+\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:
+\begin{equation}
+\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
+\mathds{B}^\mathsf{N},
+\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}
+\end{equation}
+\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
+(S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
+\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function} 
+$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$. 
+Then the general chaotic iterations defined in Equation \ref{general CIs} can 
+be described by the following discrete dynamical system:
+\begin{equation}
+\left\{
+\begin{array}{l}
+X^0 \in \mathcal{X} \\
+X^{k+1}=G_{f}(X^k).%
+\end{array}%
+\right.
+\end{equation}%

 
 

-\begin{proof}
-Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
+Another time, a shift function appears as a component of these general chaotic 
+iterations. 

 
 

-Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in \tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) = \varnothing$. Consequently, $f$ is $\tau-$transitive.
+To study the Devaney's chaos property, a distance between two points 
+$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
+Let us introduce:
+\begin{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}
+where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
+$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.

 
 

-Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a periodic point for $f$ into $V$.

 
 

-Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
+\begin{proposition}
+The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
+\end{proposition}

 
 

-But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in \mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is proven.
+\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.
+ \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 
+$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
+ \item $d_s$ is symmetric 
+($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
+of the symmetric difference. 
+\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, 
+and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, 
+we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
+inequality is obtained.
+ \end{itemize}

 \end{proof}
 
 \end{proof}
 

-\subsection{A given system can always be claimed as chaotic}

 
 

-Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point. Then this function is chaotic (in a certain way):
+Before being able to study the topological behavior of the general 
+chaotic iterations, we must firstly establish that:

 
 

-\begin{theorem}
-Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having at least a fixed point.
-Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete) topology on $\X$.
-\end{theorem}
+\begin{proposition}
+ For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 
+$\left( \mathcal{X},d\right)$.
+\end{proposition}

 
 
 \begin{proof}
 
 
 \begin{proof}

-$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus \{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq \varnothing$.
-As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For instance, $n=0$ is appropriate.
-
-Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V = \mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is regular, and the result is established.
+We use the sequential continuity.
+Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
+\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
+G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
+G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
+thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
+sequences).\newline
+As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
+to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
+d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
+In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
+cell will change its state:
+$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
+
+In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
+\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
+n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
+first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
+
+Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
+identical and strategies $S^n$ and $S$ start with the same first term.\newline
+Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
+so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
+\noindent We now prove that the distance between $\left(
+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
+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
+\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
+\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
+\begin{equation*}
+\exists n_{2}\in \mathds{N},\forall n\geqslant
+n_{2},d_{s}(S^n,S)<10^{-(k+2)},
+\end{equation*}%
+thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
+\end{itemize}
+\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
+In conclusion,
+$$
+\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
+,\forall n\geqslant N_{0},
+ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+\leqslant \varepsilon .
+$$
+$G_{f}$ is consequently continuous.

 \end{proof}
 
 
 \end{proof}
 
 

-
-
-\subsection{A given system can always be claimed as non-chaotic}
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,

 
 \begin{theorem}
 
 \begin{theorem}

-Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
-If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic (for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.

 \end{theorem}
 
 \end{theorem}
 

-\begin{proof}
-Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty}, f\right)$ is both transitive and regular.
-
-Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty}, f\right)$ is regular. Then $x$ must be a periodic point of $f$.
+Let us firstly prove the following lemma.

 
 

-Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite because  $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in \mathcal{X}, y \notin I_x$.
+\begin{lemma}[Strong transitivity]
+\label{strongTrans}
+ For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
+find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+\end{lemma}

 
 

-As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq \varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x \Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
+\begin{proof}
+ Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
+Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
+are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
+$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+the form $(S',E')$ where $E'=E$ and $S'$ starts with 
+$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+\begin{itemize}
+ \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+ \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+\end{itemize}
+Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
+where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+claimed in the lemma.

 \end{proof}
 
 \end{proof}
 

+We can now prove the Theorem~\ref{t:chaos des general}...
+
+\begin{proof}[Theorem~\ref{t:chaos des general}]
+Firstly, strong transitivity implies transitivity.
+
+Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+prove that $G_f$ is regular, it is sufficient to prove that
+there exists a strategy $\tilde S$ such that the distance between
+$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+$(\tilde S,E)$ is a periodic point.
+
+Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
+and $t_2\in\mathds{N}$ such
+that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+of $S$ and the first $t_2$ terms of $S'$: $$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+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
+have $d((S,E),(\tilde S,E))<\epsilon$.
+\end{proof}

 
 
 
 
 
 

+\section{Efficient PRNG based on Chaotic Iterations}
+\label{sec:efficient prng}

 
 

+In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
+possible to improve  previous works [ref]. One solution  consists in considering
+that the  strategy used contains all the  bits for which the  negation is
+achieved out. Then in order to apply  the negation on these bits we can simply
+apply the  xor operator between  the current number  and the strategy. In
+order to obtain the strategy we also use a classical PRNG.

 
 

-\section{Chaos on the order topology}
-
-\subsection{The phase space is an interval of the real line}
-
-\subsubsection{Toward a topological semiconjugacy}
-
-In what follows, our intention is to establish, by using a topological semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as iterations on a real interval. To do so, we must firstly introduce some notations and terminologies. 
+Here  is an  example with  16-bits numbers  showing how  the bitwise  operations
+are
+applied.  Suppose  that $x$ and the  strategy $S^i$ are defined  in binary mode.
+Then the following table shows the result of $x$ xor $S^i$.
+$$
+\begin{array}{|cc|cccccccccccccccc|}
+\hline
+x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+\hline
+S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+\hline
+x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+\hline
+
+\hline
+ \end{array}
+$$

 
 

-Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket 1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N} \times \B^\mathsf{N}$.

 
 
 
 

-\begin{definition}
-The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[ 0, 2^{10} \big[$ is defined by:
-$$
-\begin{array}{cccl}
-\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}& \longrightarrow & \big[ 0, 2^{10} \big[ \\
- & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto & \varphi \left((S,E)\right)
-\end{array}
-$$
-\noindent where $\varphi\left((S,E)\right)$ is the real number:
-\begin{itemize}
-\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$.
-\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots = \sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ 
-\end{itemize}
-\end{definition}

 
 
 
 

+\lstset{language=C,caption={C code of the sequential chaotic iterations based
+PRNG},label=algo:seqCIprng}
+\begin{lstlisting}
+unsigned int CIprng() {
+  static unsigned int x = 123123123;
+  unsigned long t1 = xorshift();
+  unsigned long t2 = xor128();
+  unsigned long t3 = xorwow();
+  x = x^(unsigned int)t1;
+  x = x^(unsigned int)(t2>>32);
+  x = x^(unsigned int)(t3>>32);
+  x = x^(unsigned int)t2;
+  x = x^(unsigned int)(t1>>32);
+  x = x^(unsigned int)t3;
+  return x;
+}
+\end{lstlisting}

 
 

-$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic iterations $\Go$ on this real interval. To do so, two intermediate functions over $\big[ 0, 2^{10} \big[$ must be introduced:

 
 
 
 

-\begin{definition}
-\label{def:e et s}
-Let $x \in \big[ 0, 2^{10} \big[$ and:
-\begin{itemize}
-\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$: $\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$.
-\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal decomposition of $x$ is the one that does not have an infinite number of 9: 
-$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$.
-\end{itemize}
-$e$ and $s$ are thus defined as follows:
-$$
-\begin{array}{cccl}
-e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\
- & x & \longmapsto & (e_0, \hdots, e_9)
-\end{array}
-$$
-\noindent and
-$$
-\begin{array}{cccl}
-s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 \rrbracket^{\mathds{N}} \\
- & x & \longmapsto & (s^k)_{k \in \mathds{N}}
-\end{array}
-$$
-\end{definition}

 
 

-We are now able to define the function $g$, whose goal is to translate the chaotic iterations $\Go$ on an interval of $\mathds{R}$.

 
 

-\begin{definition}
-$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by:
-$$
-\begin{array}{cccl}
-g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\
-& \\
- & x & \longmapsto & g(x)
-\end{array}
-$$
-\noindent where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow:
-\begin{itemize}
-\item its integral part has a binary decomposition equal to $e_0', \hdots, e_9'$, with:
-$$
-e_i' = \left\{
-\begin{array}{ll}
-e(x)_i & \textrm{ if } i \neq s^0\\
-e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\
-\end{array}
-\right.
-$$
-\item whose decimal part is $s(x)^1, s(x)^2, \hdots$
-\end{itemize}
-\end{definition}
+In listing~\ref{algo:seqCIprng}  a sequential version of  our chaotic iterations
+based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
+This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
+\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
+xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
+each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$32)}  selects the 32 most significants  bits of the
+variable \texttt{t}.   So to produce a  random number realizes  6 xor operations
+with 6 32-bits  numbers produced by 3 64-bits PRNG.   This version successes the
+BigCrush of the TestU01 battery~\cite{LEcuyerS07}.

 
 

-\bigskip
+\section{Efficient PRNGs based on chaotic iterations on GPU}
+\label{sec:efficient prng gpu}

 
 

+In  order to benefit  from computing  power of  GPU, a  program needs  to define
+independent blocks of threads which  can be computed simultaneously. In general,
+the larger the number of threads is,  the more local memory is used and the less
+branching  instructions are  used (if,  while, ...),  the better  performance is
+obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
+previous section, it is possible to  build a similar program which computes PRNG
+on   GPU.  In  the   CUDA~\cite{Nvid10}  environment,   threads  have   a  local
+identificator, called \texttt{ThreadIdx} relative to the block containing them.
+
+
+\subsection{Naive version for GPU}
+
+From the CPU version, it is possible  to obtain a quite similar version for GPU.
+The principe consists in assigning the computation of a PRNG as in sequential to
+each thread  of the  GPU.  Of course,  it is  essential that the  three xor-like
+PRNGs  used for  our computation  have different  parameters. So  we  chose them
+randomly with  another PRNG. As the  initialisation is performed by  the CPU, we
+have  chosen  to  use  the  ISAAC  PRNG~\cite{Jenkins96}  to  initalize  all  the
+parameters for  the GPU version  of our PRNG.   The implementation of  the three
+xor-like  PRNGs  is  straightforward  as  soon as  their  parameters  have  been
+allocated in  the GPU memory.  Each xor-like PRNGs  used works with  an internal
+number  $x$  which keeps  the  last  generated  random numbers.  Other  internal
+variables  are   also  used   by  the  xor-like   PRNGs.  More   precisely,  the
+implementation of the  xor128, the xorshift and the  xorwow respectively require
+4, 5 and 6 unsigned long as internal variables.

 
 

-In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k +  \sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then: $$g(x) = \displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) +  \sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}.$$
+\begin{algorithm}

 
 

-\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}
+\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
+PRNGs in global memory\;
+NumThreads: Number of threads\;}
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadIdx is concerned by the computation} {
+  retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
+  \For{i=1 to n} {
+    compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
+    store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
+  }
+  store internal variables in InternalVarXorLikeArray[threadIdx]\;
+}

 
 

-Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most usual one being the Euclidian distance recalled bellow:
+\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
+\label{algo:gpu_kernel}
+\end{algorithm}

 
 

-\begin{notation}
-\index{distance!euclidienne}
-$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is, $\Delta(x,y) = |y-x|^2$.
-\end{notation}
+Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of  PRNG using
+GPU.  According  to the available  memory in the  GPU and the number  of threads
+used simultenaously,  the number  of random numbers  that a thread  can generate
+inside   a    kernel   is   limited,   i.e.    the    variable   \texttt{n}   in
+algorithm~\ref{algo:gpu_kernel}. For example, if  $100,000$ threads are used and
+if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}
+then   the  memory   required   to  store   internals   variables  of   xor-like
+PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
+and  random  number of  our  PRNG  is  equals to  $100,000\times  ((4+5+6)\times
+2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb.
+
+All the  tests performed  to pass the  BigCrush of TestU01  succeeded. Different
+number of threads, called \texttt{NumThreads} in our algorithm, have been tested
+upto $10$ millions.
+\newline
+\newline
+{\bf QUESTION : on laisse cette remarque, je suis mitigé !!!}

 
 

-\medskip
+\begin{remark}
+Algorithm~\ref{algo:gpu_kernel}  has  the  advantage to  manipulate  independent
+PRNGs, so this version is easily usable on a cluster of computer. The only thing
+to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in
+using a master node for the initialization which computes the initial parameters
+for all the differents nodes involves in the computation.
+\end{remark}

 
 

-This Euclidian distance does not reproduce exactly the notion of proximity induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$. This is the reason why we have to introduce the following metric:
+\subsection{Improved version for GPU}

 
 

+As GPU cards using CUDA have shared memory between threads of the same block, it
+is possible  to use this  feature in order  to simplify the  previous algorithm,
+i.e., using less  than 3 xor-like PRNGs. The solution  consists in computing only
+one xor-like PRNG by thread, saving  it into shared memory and using the results
+of some  other threads in the  same block of  threads. In order to  define which
+thread uses the result of which other  one, we can use a permutation array which
+contains  the indexes  of  all threads  and  for which  a  permutation has  been
+performed.  In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used.
+The    variable   \texttt{offset}    is    computed   using    the   value    of
+\texttt{permutation\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}
+which represent the indexes of the  other threads for which the results are used
+by the  current thread. In  the algorithm, we  consider that a  64-bits xor-like
+PRNG is used, that is why both 32-bits parts are used.

 
 

+This version also succeeds to the {\it BigCrush} batteries of tests.

 
 

-\begin{definition}
-Let $x,y \in \big[ 0, 2^{10} \big[$.
-$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$ defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$, where:
-\begin{center}
-$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k, \check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$.
-\end{center}
-\end{definition}
+\begin{algorithm}

 
 

-\begin{proposition}
-$D$ is a distance on $\big[ 0, 2^{10} \big[$.
-\end{proposition}
+\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
+in global memory\;
+NumThreads: Number of threads\;
+tab1, tab2: Arrays containing permutations of size permutation\_size\;}

 
 

-\begin{proof}
-The three axioms defining a distance must be checked.
-\begin{itemize}
-\item $D \geqslant 0$, because everything is positive in its definition. If $D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal (they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then $\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$.
-\item $D(x,y)=D(y,x)$.
-\item Finally, the triangular inequality is obtained due to the fact that both $\delta$ and $\Delta(x,y)=|x-y|$ satisfy it.
-\end{itemize}
-\end{proof}
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadId is concerned} {
+  retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory\;
+  offset = threadIdx\%permutation\_size\;
+  o1 = threadIdx-offset+tab1[offset]\;
+  o2 = threadIdx-offset+tab2[offset]\;
+  \For{i=1 to n} {
+    t=xor-like()\;
+    t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\;
+    shared\_mem[threadId]=t\;
+    x = x $\oplus$ t\;

 
 

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

 
 

-The convergence of sequences according to $D$ is not the same than the usual convergence related to the Euclidian metric. For instance, if $x^n \to x$ according to $D$, then necessarily the integral part of each $x^n$ is equal to the integral part of $x$ (at least after a given threshold), and the decimal part of $x^n$ corresponds to the one of $x$ ``as far as required''.
-To illustrate this fact, a comparison between $D$ and the Euclidian distance is given Figure \ref{fig:comparaison de distances}. These illustrations show that $D$ is richer and more refined than the Euclidian distance, and thus is more precise.
+\caption{main kernel for the chaotic iterations based PRNG GPU efficient
+version}
+\label{algo:gpu_kernel2}
+\end{algorithm}

 
 

+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in three operations having 
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, three iterations of the general chaotic
+iterations are realized between two stored values of the PRNG.
+To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+we must guarantee that this dynamical system iterates on the space 
+$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
+The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+To prevent from any flaws of chaotic properties, we must check that each right 
+term, corresponding to terms of the strategies,  can possibly be equal to any
+integer of $\llbracket 1, \mathsf{N} \rrbracket$. 
+
+Such a result is obvious for the two first lines, as for the xor-like(), all the
+integers belonging into its interval of definition can occur at each iteration.
+It can be easily stated for the two last lines by an immediate mathematical
+induction.
+
+Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
+chaotic iterations presented previously, and for this reason, it satisfies the 
+Devaney's formulation of a chaotic behavior.

 
 

-\begin{figure}[t]
+\section{Experiments}
+\label{sec:experiments}
+
+Different experiments  have been  performed in order  to measure  the generation
+speed. We have used  a computer equiped with Tesla C1060 NVidia  GPU card and an
+Intel  Xeon E5530 cadenced  at 2.40  GHz for  our experiments  and we  have used
+another one  equipped with  a less performant  CPU and  a GeForce GTX  280. Both
+cards have 240 cores.
+
+In Figure~\ref{fig:time_gpu}  we compare the number of  random numbers generated
+per second. The xor-like prng  is a xor64 described in~\cite{Marsaglia2003}.  In
+order to obtain the optimal performance  we remove the storage of random numbers
+in the GPU memory. This step is time consumming and slows down the random number
+generation.  Moreover, if you are interested by applications that consome random
+numbers  directly   when  they  are  generated,  their   storage  is  completely
+useless. In this  figure we can see  that when the number of  threads is greater
+than approximately 30,000 upto 5 millions the number of random numbers generated
+per second  is almost constant.  With the  naive version, it is  between 2.5 and
+3GSample/s.   With  the  optimized   version,  it  is  approximately  equals  to
+20GSample/s. Finally  we can remark  that both GPU  cards are quite  similar. In
+practice,  the Tesla C1060  has more  memory than  the GTX  280 and  this memory
+should be of better quality.
+
+\begin{figure}[htbp]

 \begin{center}
 \begin{center}

-  \subfigure[Function $x \to dist(x;1,234) $ on the interval $(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad
-  \subfigure[Function $x \to dist(x;3) $ on the interval $(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}}
+  \includegraphics[scale=.7]{curve_time_gpu.pdf}

 \end{center}
 \end{center}

-\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).}
-\label{fig:comparaison de distances}
+\caption{Number of random numbers generated per second}
+\label{fig:time_gpu}

 \end{figure}
 
 
 \end{figure}
 
 

+In  comparison,   Listing~\ref{algo:seqCIprng}  allows  us   to  generate  about
+138MSample/s with only one core of the Xeon E5530.

 
 
 
 

-\subsubsection{The semiconjugacy}

 
 

-It is now possible to define a topological semiconjugacy between $\mathcal{X}$ and an interval of $\mathds{R}$:

 
 

-\begin{theorem}
-Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on $\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow:
-\begin{equation*}
-\begin{CD}
-\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>> \left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\
-    @V{\varphi}VV                    @VV{\varphi}V\\
-\left( ~\big[ 0, 2^{10} \big[, D~\right)  @>>g> \left(~\big[ 0, 2^{10} \big[, D~\right)
-\end{CD}
-\end{equation*}
-\end{theorem}

 
 

-\begin{proof}
-$\varphi$ has been constructed in order to be continuous and onto.
-\end{proof}
+%% \section{Cryptanalysis of the Proposed PRNG}

 
 

-In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} \big[$.

 
 

+%% Mettre ici la preuve de PCH

 
 

+%\section{The relativity of disorder}
+%\label{sec:de la relativité du désordre}

 
 

+%In the next two sections, we investigate the impact of the choices that have
+%lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.

 
 

+%\subsection{Impact of the topology's finenesse}

 
 

+%Let us firstly introduce the following notations.

 
 

-\subsection{Study of the chaotic iterations described as a real function}
+%\begin{notation}
+%$\mathcal{X}_\tau$ will denote the topological space
+%$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
+%of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
+%$\mathcal{V} (x)$, if there is no ambiguity).
+%\end{notation}

 
 
 
 

-\begin{figure}[t]
-\begin{center}
-  \subfigure[ICs on the interval $(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad
-  \subfigure[ICs on the interval $(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\
-  \subfigure[ICs on the interval $(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad
-  \subfigure[ICs on the interval $(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}}
-\end{center}
-\caption{Representation of the chaotic iterations.}
-\label{fig:ICs}
-\end{figure}

 
 

+%\begin{theorem}
+%\label{Th:chaos et finesse}
+%Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t.
+%$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous
+%both for $\tau$ and $\tau'$.

 
 

+%If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then
+%$(\mathcal{X}_\tau,f)$ is chaotic too.
+%\end{theorem}

 
 

+%\begin{proof}
+%Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.

 
 

-\begin{figure}[t]
-\begin{center}
-  \subfigure[ICs on the interval $(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad
-  \subfigure[ICs on the interval $(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}}
-\end{center}
-\caption{ICs on small intervals.}
-\label{fig:ICs2}
-\end{figure}
+%Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in
+%\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we
+%can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) =
+%\varnothing$. Consequently, $f$ is $\tau-$transitive.

 
 

-\begin{figure}[t]
-\begin{center}
-  \subfigure[ICs on the interval $(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad
-  \subfigure[ICs on the interval  $(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad
-\end{center}
-\caption{General aspect of the chaotic iterations.}
-\label{fig:ICs3}
-\end{figure}
+%Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for
+%all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a
+%periodic point for $f$ into $V$.

 
 

+%Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood
+%of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.

 
 

-We have written a Python program to represent the chaotic iterations with the vectorial negation on the real line $\mathds{R}$. Various representations of these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}. It can be remarked that the function $g$ is a piecewise linear function: it is linear on each interval having the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its slope is equal to 10. Let us justify these claims:
+%But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in
+%\mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a
+%periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is
+%proven. 
+%\end{proof}

 
 

-\begin{proposition}
-\label{Prop:derivabilite des ICs}
-Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on $\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
+%\subsection{A given system can always be claimed as chaotic}

 
 

-Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, $g$ is a linear function, having a slope equal to 10: $\forall x \notin I, g'(x)=10$.
-\end{proposition}
+%Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
+%Then this function is chaotic (in a certain way):

 
 

+%\begin{theorem}
+%Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having
+%at least a fixed point.
+%Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete)
+%topology on $\X$.
+%\end{theorem}

 
 

-\begin{proof}
-Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral prat $e$ and the same decimal part $s^0$: on the set $I_n$,  functions $e(x)$ and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all the images $g(x)$ of these points $x$:
-\begin{itemize}
-\item Have the same integral part, which is $e$, except probably the bit number $s^0$. In other words, this integer has approximately the same binary decomposition than $e$, the sole exception being the digit $s^0$ (this number is then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$, \emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$).
-\item A shift to the left has been applied to the decimal part $y$, losing by doing so the common first digit $s^0$. In other words, $y$ has been mapped into $10\times y - s^0$.
-\end{itemize}
-To sum up, the action of $g$ on the points of $I$ is as follows: first, make a multiplication by 10, and second, add the same constant to each term, which is $\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$.
-\end{proof}

 
 

-\begin{remark}
-Finally, chaotic iterations are elements of the large family of functions that are both chaotic and piecewise linear (like the tent map).
-\end{remark}
+%\begin{proof}
+%$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus
+%\{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq
+%\varnothing$.
+%As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for
+%an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For
+%instance, $n=0$ is appropriate.

 
 

+%Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V =
+%\mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is
+%regular, and the result is established.
+%\end{proof}

 
 
 
 

-\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}

 
 

-The two propositions bellow allow to compare our two distances on $\big[ 0, 2^\mathsf{N} \big[$:

 
 

-\begin{proposition}
-Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0, 2^\mathsf{N} \big[, D~\right)$ is not continuous. 
-\end{proposition}
+%\subsection{A given system can always be claimed as non-chaotic}

 
 

-\begin{proof}
-The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is such that:
-\begin{itemize}
-\item $\Delta (x^n,2) \to 0.$
-\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0.
-\end{itemize}
+%\begin{theorem}
+%Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
+%If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic
+%(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
+%\end{theorem}

 
 

-The sequential characterization of the continuity concludes the demonstration.
-\end{proof}
+%\begin{proof}
+%Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
+%f\right)$ is both transitive and regular.

 
 

+%Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must
+%contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty},
+%f\right)$ is regular. Then $x$ must be a periodic point of $f$.

 
 

+%Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite
+%because  $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in
+%\mathcal{X}, y \notin I_x$.
+
+%As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty
+%sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq
+%\varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x
+%\Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
+%\end{proof}
+
+
+
+
+
+
+%\section{Chaos on the order topology}
+%\label{sec: chaos order topology}
+%\subsection{The phase space is an interval of the real line}
+
+%\subsubsection{Toward a topological semiconjugacy}
+
+%In what follows, our intention is to establish, by using a topological
+%semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as
+%iterations on a real interval. To do so, we must firstly introduce some
+%notations and terminologies. 
+
+%Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket
+%1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N}
+%\times \B^\mathsf{N}$.
+
+
+%\begin{definition}
+%The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[
+%0, 2^{10} \big[$ is defined by:
+%\begin{equation}
+% \begin{array}{cccl}
+%\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}&
+%\longrightarrow & \big[ 0, 2^{10} \big[ \\
+% & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto &
+%\varphi \left((S,E)\right)
+%\end{array}
+%\end{equation}
+%where $\varphi\left((S,E)\right)$ is the real number:
+%\begin{itemize}
+%\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that
+%is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$.
+%\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots =
+%\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ 
+%\end{itemize}
+%\end{definition}
+
+
+
+%$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a
+%real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic
+%iterations $\Go$ on this real interval. To do so, two intermediate functions
+%over $\big[ 0, 2^{10} \big[$ must be introduced:
+
+
+%\begin{definition}
+%\label{def:e et s}
+%Let $x \in \big[ 0, 2^{10} \big[$ and:
+%\begin{itemize}
+%\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$:
+%$\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$.
+%\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal
+%decomposition of $x$ is the one that does not have an infinite number of 9: 
+%$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$.
+%\end{itemize}
+%$e$ and $s$ are thus defined as follows:
+%\begin{equation}
+%\begin{array}{cccl}
+%e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\
+% & x & \longmapsto & (e_0, \hdots, e_9)
+%\end{array}
+%\end{equation}
+%and
+%\begin{equation}
+% \begin{array}{cccc}
+%s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9
+%\rrbracket^{\mathds{N}} \\
+% & x & \longmapsto & (s^k)_{k \in \mathds{N}}
+%\end{array}
+%\end{equation}
+%\end{definition}
+
+%We are now able to define the function $g$, whose goal is to translate the
+%chaotic iterations $\Go$ on an interval of $\mathds{R}$.
+
+%\begin{definition}
+%$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by:
+%\begin{equation}
+%\begin{array}{cccc}
+%g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\
+% & x & \longmapsto & g(x)
+%\end{array}
+%\end{equation}
+%where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow:
+%\begin{itemize}
+%\item its integral part has a binary decomposition equal to $e_0', \hdots,
+%e_9'$, with:
+% \begin{equation}
+%e_i' = \left\{
+%\begin{array}{ll}
+%e(x)_i & \textrm{ if } i \neq s^0\\
+%e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\
+%\end{array}
+%\right.
+%\end{equation}
+%\item whose decimal part is $s(x)^1, s(x)^2, \hdots$
+%\end{itemize}
+%\end{definition}
+
+%\bigskip
+
+
+%In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k + 
+%\sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then:
+%\begin{equation}
+%g(x) =
+%\displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + 
+%\sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}. 
+%\end{equation}
+
+
+%\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}
+
+%Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most
+%usual one being the Euclidian distance recalled bellow:
+
+%\begin{notation}
+%\index{distance!euclidienne}
+%$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is,
+%$\Delta(x,y) = |y-x|^2$.
+%\end{notation}
+
+%\medskip
+
+%This Euclidian distance does not reproduce exactly the notion of proximity
+%induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$.
+%This is the reason why we have to introduce the following metric:
+
+
+
+%\begin{definition}
+%Let $x,y \in \big[ 0, 2^{10} \big[$.
+%$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$
+%defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$,
+%where:
+%\begin{center}
+%$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k,
+%\check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty
+%\dfrac{|S^k-\check{S}^k|}{10^k}}$.
+%\end{center}
+%\end{definition}
+
+%\begin{proposition}
+%$D$ is a distance on $\big[ 0, 2^{10} \big[$.
+%\end{proposition}
+
+%\begin{proof}
+%The three axioms defining a distance must be checked.
+%\begin{itemize}
+%\item $D \geqslant 0$, because everything is positive in its definition. If
+%$D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal
+%(they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then
+%$\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have
+%the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$.
+%\item $D(x,y)=D(y,x)$.
+%\item Finally, the triangular inequality is obtained due to the fact that both
+%$\delta$ and $\Delta(x,y)=|x-y|$ satisfy it.
+%\end{itemize}
+%\end{proof}
+
+
+%The convergence of sequences according to $D$ is not the same than the usual
+%convergence related to the Euclidian metric. For instance, if $x^n \to x$
+%according to $D$, then necessarily the integral part of each $x^n$ is equal to
+%the integral part of $x$ (at least after a given threshold), and the decimal
+%part of $x^n$ corresponds to the one of $x$ ``as far as required''.
+%To illustrate this fact, a comparison between $D$ and the Euclidian distance is
+%given Figure \ref{fig:comparaison de distances}. These illustrations show that
+%$D$ is richer and more refined than the Euclidian distance, and thus is more
+%precise.
+
+
+%\begin{figure}[t]
+%\begin{center}
+%  \subfigure[Function $x \to dist(x;1,234) $ on the interval
+%$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad
+%  \subfigure[Function $x \to dist(x;3) $ on the interval
+%$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}}
+%\end{center}
+%\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).}
+%\label{fig:comparaison de distances}
+%\end{figure}
+
+
+
+
+%\subsubsection{The semiconjugacy}
+
+%It is now possible to define a topological semiconjugacy between $\mathcal{X}$
+%and an interval of $\mathds{R}$:
+
+%\begin{theorem}
+%Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on
+%$\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow:
+%\begin{equation*}
+%\begin{CD}
+%\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>>
+%\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\
+%    @V{\varphi}VV                    @VV{\varphi}V\\
+%\left( ~\big[ 0, 2^{10} \big[, D~\right)  @>>g> \left(~\big[ 0, 2^{10} \big[,
+%D~\right)
+%\end{CD}
+%\end{equation*}
+%\end{theorem}
+
+%\begin{proof}
+%$\varphi$ has been constructed in order to be continuous and onto.
+%\end{proof}
+
+%In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N}
+%\big[$.
+
+
+
+
+
+
+%\subsection{Study of the chaotic iterations described as a real function}
+
+
+%\begin{figure}[t]
+%\begin{center}
+%  \subfigure[ICs on the interval
+%$(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad
+%  \subfigure[ICs on the interval
+%$(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\
+%  \subfigure[ICs on the interval
+%$(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad
+%  \subfigure[ICs on the interval
+%$(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}}
+%\end{center}
+%\caption{Representation of the chaotic iterations.}
+%\label{fig:ICs}
+%\end{figure}
+
+
+
+
+%\begin{figure}[t]
+%\begin{center}
+%  \subfigure[ICs on the interval
+%$(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad
+%  \subfigure[ICs on the interval
+%$(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}}
+%\end{center}
+%\caption{ICs on small intervals.}
+%\label{fig:ICs2}
+%\end{figure}
+
+%\begin{figure}[t]
+%\begin{center}
+%  \subfigure[ICs on the interval
+%$(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad
+%  \subfigure[ICs on the interval 
+%$(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad
+%\end{center}
+%\caption{General aspect of the chaotic iterations.}
+%\label{fig:ICs3}
+%\end{figure}
+
+
+%We have written a Python program to represent the chaotic iterations with the
+%vectorial negation on the real line $\mathds{R}$. Various representations of
+%these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}.
+%It can be remarked that the function $g$ is a piecewise linear function: it is
+%linear on each interval having the form $\left[ \dfrac{n}{10},
+%\dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its
+%slope is equal to 10. Let us justify these claims:
+
+%\begin{proposition}
+%\label{Prop:derivabilite des ICs}
+%Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on
+%$\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{
+%\dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
+
+%Furthermore, on each interval of the form $\left[ \dfrac{n}{10},
+%\dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$,
+%$g$ is a linear function, having a slope equal to 10: $\forall x \notin I,
+%g'(x)=10$.
+%\end{proposition}

 
 

-A contrario:

 
 

-\begin{proposition}
-Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0, 2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction. 
-\end{proposition}
+%\begin{proof}
+%Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket
+%0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral
+%prat $e$ and the same decimal part $s^0$: on the set $I_n$,  functions $e(x)$
+%and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all
+%the images $g(x)$ of these points $x$:
+%\begin{itemize}
+%\item Have the same integral part, which is $e$, except probably the bit number
+%$s^0$. In other words, this integer has approximately the same binary
+%decomposition than $e$, the sole exception being the digit $s^0$ (this number is
+%then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$,
+%\emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$).
+%\item A shift to the left has been applied to the decimal part $y$, losing by
+%doing so the common first digit $s^0$. In other words, $y$ has been mapped into
+%$10\times y - s^0$.
+%\end{itemize}
+%To sum up, the action of $g$ on the points of $I$ is as follows: first, make a
+%multiplication by 10, and second, add the same constant to each term, which is
+%$\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$.
+%\end{proof}
+
+%\begin{remark}
+%Finally, chaotic iterations are elements of the large family of functions that
+%are both chaotic and piecewise linear (like the tent map).
+%\end{remark}
+
+
+
+%\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}
+
+%The two propositions bellow allow to compare our two distances on $\big[ 0,
+%2^\mathsf{N} \big[$:
+
+%\begin{proposition}
+%Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0,
+%2^\mathsf{N} \big[, D~\right)$ is not continuous. 
+%\end{proposition}
+
+%\begin{proof}
+%The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is
+%such that:
+%\begin{itemize}
+%\item $\Delta (x^n,2) \to 0.$
+%\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0.
+%\end{itemize}
+
+%The sequential characterization of the continuity concludes the demonstration.
+%\end{proof}
+
+
+
+%A contrario:
+
+%\begin{proposition}
+%Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0,
+%2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction. 
+%\end{proposition}
+
+%\begin{proof}
+%If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given
+%threshold, because $D_e$ only returns integers. So, after this threshold, the
+%integral parts of all the $x^n$ are equal to the integral part of $x$. 
+
+%Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k
+%\in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This
+%means that for all $k$, an index $N_k$ can be found such that, $\forall n
+%\geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the
+%digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the
+%result.
+%\end{proof}
+
+%The conclusion of these propositions is that the proposed metric is more precise
+%than the Euclidian distance, that is:

 
 

-\begin{proof}
-If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given threshold, because $D_e$ only returns integers. So, after this threshold, the integral parts of all the $x^n$ are equal to the integral part of $x$. 
+%\begin{corollary}
+%$D$ is finer than the Euclidian distance $\Delta$.
+%\end{corollary}

 
 

-Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k \in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This means that for all $k$, an index $N_k$ can be found such that, $\forall n \geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the result.
-\end{proof}
+%This corollary can be reformulated as follows:
+
+%\begin{itemize}
+%\item The topology produced by $\Delta$ is a subset of the topology produced by
+%$D$.
+%\item $D$ has more open sets than $\Delta$.
+%\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than
+%to converge with the one inherited by $\Delta$, which is denoted here by
+%$\tau_\Delta$.
+%\end{itemize}

 
 

-The conclusion of these propositions is that the proposed metric is more precise than the Euclidian distance, that is:

 
 

-\begin{corollary}
-$D$ is finer than the Euclidian distance $\Delta$.
-\end{corollary}
+%\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
+%\label{chpt:Chaos des itérations chaotiques sur R}

 
 

-This corollary can be reformulated as follows:

 
 

-\begin{itemize}
-\item The topology produced by $\Delta$ is a subset of the topology produced by $D$.
-\item $D$ has more open sets than $\Delta$.
-\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than to converge with the one inherited by $\Delta$, which is denoted here by $\tau_\Delta$.
-\end{itemize}

 
 

+%\subsubsection{Chaos according to Devaney}

 
 

-\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
-\label{chpt:Chaos des itérations chaotiques sur R}
+%We have recalled previously that the chaotic iterations $\left(\Go,
+%\mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We
+%can deduce that they are chaotic on $\mathds{R}$ too, when considering the order
+%topology, because:
+%\begin{itemize}
+%\item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10}
+%\big[_D\right)$ are semiconjugate by $\varphi$,
+%\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic
+%according to Devaney, because the semiconjugacy preserve this character.
+%\item But the topology generated by $D$ is finer than the topology generated by
+%the Euclidian distance $\Delta$ -- which is the order topology.
+%\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the
+%chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order
+%topology on $\mathds{R}$.
+%\end{itemize}

 
 

+%This result can be formulated as follows.

 
 

+%\begin{theorem}
+%\label{th:IC et topologie de l'ordre}
+%The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the
+%Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the
+%order topology.
+%\end{theorem}

 
 

-\subsubsection{Chaos according to Devaney}
+%Indeed this result is weaker than the theorem establishing the chaos for the
+%finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre}
+%still remains important. Indeed, we have studied in our previous works a set
+%different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$),
+%in order to be as close as possible from the computer: the properties of
+%disorder proved theoretically will then be preserved when computing. However, we
+%could wonder whether this change does not lead to a disorder of a lower quality.
+%In other words, have we replaced a situation of a good disorder lost when
+%computing, to another situation of a disorder preserved but of bad quality.
+%Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
+% 

 
 

-We have recalled previously that the chaotic iterations $\left(\Go, \mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We can deduce that they are chaotic on $\mathds{R}$ too, when considering the order topology, because:
-\begin{itemize}
-\item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10} \big[_D\right)$ are semiconjugate by $\varphi$,
-\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic according to Devaney, because the semiconjugacy preserve this character.
-\item But the topology generated by $D$ is finer than the topology generated by the Euclidian distance $\Delta$ -- which is the order topology.
-\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order topology on $\mathds{R}$.
-\end{itemize}

 
 

-This result can be formulated as follows.

 
 

-\begin{theorem}
-\label{th:IC et topologie de l'ordre}
-The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the order topology.
-\end{theorem}

 
 

-Indeed this result is weaker than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
- 

 
 
 
 

+\section{Security Analysis}

 
 

-\section{Efficient prng based on chaotic iterations}

 
 

-On parle du séquentiel avec des nombres 64 bits\\

 
 
 
 

-In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
-possible to improve  previous works [ref]. One solution  consists in considering
-that the  strategy used contains all the  bits for which the  negation is
-achieved out. Then instead of applying  the negation on these bits we can simply
-apply the  xor operator between  the current number  and the strategy. In
-order to obtain the strategy we also use a classical PRNG.
+In this section the concatenation of two strings $u$ and $v$ is classically
+denoted by $uv$.
+In a cryptographic context, a pseudo-random generator is a deterministic
+algorithm $G$ transforming strings  into strings and such that, for any
+seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
+$\ell_G(N)$ with $\ell_G(N)>N$.
+The notion of {\it secure} PRNGs can now be defined as follows. 

 
 

-%% \begin{figure}[htbp]
-%% \begin{center}
-%% \fbox{
-%% \begin{minipage}{14cm}
-%% unsigned int CIprng() \{\\
-%%   static unsigned int x = 123123123;\\
-%%   unsigned long t1 = xorshift();\\
-%%   unsigned long t2 = xor128();\\
-%%   unsigned long t3 = xorwow();\\
-%%   x = x\textasciicircum (unsigned int)t1;\\
-%%   x = x\textasciicircum (unsigned int)(t2$>>$32);\\
-%%   x = x\textasciicircum (unsigned int)(t3$>>$32);\\
-%%   x = x\textasciicircum (unsigned int)t2;\\
-%%   x = x\textasciicircum (unsigned int)(t1$>>$32);\\
-%%   x = x\textasciicircum (unsigned int)t3;\\
-%%   return x;\\
-%% \}
-%% \end{minipage}
-%% }
-%% \end{center}
-%% \caption{sequential Chaotic Iteration PRNG}
-%% \label{algo:seqCIprng}
-%% \end{figure}
-
-
-
-\lstset{language=C,caption={C code of the sequential chaotic iterations based PRNG},label=algo:seqCIprng}
-\begin{lstlisting}
-unsigned int CIprng() {
-  static unsigned int x = 123123123;
-  unsigned long t1 = xorshift();
-  unsigned long t2 = xor128();
-  unsigned long t3 = xorwow();
-  x = x^(unsigned int)t1;
-  x = x^(unsigned int)(t2>>32);
-  x = x^(unsigned int)(t3>>32);
-  x = x^(unsigned int)t2;
-  x = x^(unsigned int)(t1>>32);
-  x = x^(unsigned int)t3;
-  return x;
-}
-\end{lstlisting}
+\begin{definition}
+A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
+algorithm $D$, for any positive polynomial $p$, and for all sufficiently
+large $k$'s,
+$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)}=1]|< \frac{1}{p(N)},$$
+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
+internal coin tosses of $D$. 
+\end{definition}

 
 

+Intuitively, it means that there is no polynomial time algorithm that can
+distinguish a perfect uniform random generator from $G$ with a non
+negligible probability. The interested reader is referred
+to~\cite[chapter~3]{Goldreich} for more information. Note that it is
+quite easily possible to change the function $\ell$ into any polynomial
+function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}.
+
+The generation schema developed in (\ref{equation Oplus}) is based on a
+pseudo-random generator. Let $H$ be a cryptographic PRNG. We may assume,
+without loss of generality, that for any string $S_0$ of size $N$, the size
+of $H(S_0)$ is $kN$, with $k>2$. It means that $\ell_H(N)=kN$. 
+Let $S_1,\ldots,S_k$ be the 
+strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concatenation of
+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 $\ell_{X}(2N)=kN=\ell_H(N)$. 
+We claim now that if this PRNG is secure,
+then the new one is secure too.

 
 

+\begin{proposition}
+If $H$ is a secure cryptographic PRNG, then $X$ is a secure cryptographic
+PRNG too.
+\end{proposition}

 
 

+\begin{proof}
+The proposition is proved by contraposition. Assume that $X$ is not
+secure. By Definition, there exists a polynomial time probabilistic
+algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
+$N\geq \frac{k_0}{2}$ satisfying 
+$$| \mathrm{Pr}[D(X(U_{2N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)}.$$
+We describe a new probabilistic algorithm $D^\prime$ on an input $w$ of size
+$kN$:
+\begin{enumerate}
+\item Decompose $w$ into $w=w_1\ldots w_{k}$, where each $w_i$ has size $N$.
+\item Pick a string $y$ of size $N$ uniformly at random.
+\item Compute $z=(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y
+  \bigoplus_{i=1}^{i=k} w_i).$
+\item Return $D(z)$.
+\end{enumerate}
+
+
+Consider  for each $y\in \mathbb{B}^{kN}$ the function $\varphi_{y}$
+from $\mathbb{B}^{kN}$ into $\mathbb{B}^{kN}$ mapping $w=w_1\ldots w_k$
+(each $w_i$ has length $N$) to 
+$(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y
+  \bigoplus_{i=1}^{i=k_1} w_i).$ By construction, one has for every $w$,
+\begin{equation}\label{PCH-1}
+D^\prime(w)=D(\varphi_y(w)),
+\end{equation}
+where $y$ is randomly generated. 
+Moreover, for each $y$, $\varphi_{y}$ is injective: if 
+$(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y\bigoplus_{i=1}^{i=k_1}
+w_i)=(y\oplus w_1^\prime)(y\oplus w_1^\prime\oplus w_2^\prime)\ldots
+(y\bigoplus_{i=1}^{i=k} w_i^\prime)$, then for every $1\leq j\leq k$,
+$y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. 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
+\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].
+\end{equation}

 
 

+Now, using (\ref{PCH-1}) again, one has  for every $x$,
+\begin{equation}\label{PCH-3}
+D^\prime(H(x))=D(\varphi_y(H(x))),
+\end{equation}
+where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$,
+thus
+\begin{equation}\label{PCH-3}
+D^\prime(H(x))=D(yx),
+\end{equation}
+where $y$ is randomly generated. 
+It follows that 

 
 

-In listing~\ref{algo:seqCIprng}  a sequential  version of our  chaotic iterations
-based PRNG  is presented.  This version  uses three classical  64-bits PRNG: the
-\texttt{xorshift},   the  \texttt{xor128}  and   the  \texttt{xorwow}.   In  the
-following,  we  call them  xor-like  PRNGSs.   These  three PRNGs  are  presented
-in~\cite{Marsaglia2003}.  As each xor-like  PRNG used works with 64-bits  and as our PRNG
-works  with 32-bits, the  use of  \texttt{(unsigned int)}  selects the  32 least
-significant bits  whereas \texttt{(unsigned int)(t3$>>$32)} selects  the 32 most
-significants  bits of the  variable \texttt{t}.  So to  produce a  random number
-realizes 6  xor operations with  6 32-bits numbers  produced by 3  64-bits PRNG.
-This version  successes the  BigCrush of the  TestU01 battery [P.   L’ecuyer and
-  R. Simard. Testu01].
+\begin{equation}\label{PCH-4}
+\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
+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. 
+\end{proof}

 
 

-\section{Efficient prng based on chaotic iterations on GPU}

 
 

-In  order to benefit  from computing  power of  GPU, a  program needs  to define
-independent blocks of threads which  can be computed simultaneously. In general,
-the larger the number of threads is,  the more local memory is used and the less
-branching  instructions are  used (if,  while, ...),  the better  performance is
-obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
-previous section, it is possible to  build a similar program which computes PRNG
-on  GPU. The principe  consists in  assigning the  computation of  a PRNG  as in
-sequential to each thread of the GPU.  Of course, it is essential that the three
-xor-like PRNGs used  for our computation have different  parameters. So we chose
-them randomly with another PRNG. As  the initialisation is performed by the CPU,
-we have chosen to use the ISAAC PRNG to initalize all the parameters for the GPU
-version  of  our  PRNG.  The  implementation  of  the  three xor-like  PRNGs  is
-straightforward  as soon  as their  parameters have  been allocated  in  the GPU
-memory. Each xor-like  PRNGs used works with an internal  number $x$ which keeps
-the  last generated random  numbers. Other  internal variables  are used  by the
-xor-like PRNGs. More  precisely, the implementation of the  xor128, the xorshift
-and  the xorwow  respectively  require 4,  5  and 6  unsigned  long as  internal
-variables.

 
 

-\begin{algorithm}

 
 

-\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like PRNGs in global memory\;
-NumThreads: Number of threads\;}
-\KwOut{NewNb: array containing random numbers in global memory}
-\If{threadId is concerned} {
-  retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
-  \For{i=1 to n} {
-    compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
-    store the new PRNG in NewNb[NumThreads*threadId+i]\;
-  }
-  store internal variables in InternalVarXorLikeArray[threadId]\;
-}

 
 

-\caption{PRNG with chaotic functions}
-\label{main kernel for the chaotic iterations based PRNG GPU version}
-\end{algorithm}
+\section{Conclusion}

 
 

-\section{Experiments}

 
 

-On passe le BigCrush\\
-On donne des temps de générations sur GPU/CPU\\
-On donne des temps de générations de nombre sur GPU puis on rappatrie sur CPU / CPU ? bof bof, on verra
+In  this  paper  we have  presented  a  new  class  of  PRNGs based  on  chaotic
+iterations. We have proven that these PRNGs are chaotic in the sense of Devenay. 

 
 

+An efficient implementation on GPU allows us to generate a huge number of pseudo
+random numbers  per second  (about 20Gsample/s). Our  PRNGs succeed to  pass the
+hardest batteries of test (TestU01).
+
+In future  work we plan  to extend our  work in order to  have cryptographically
+secure PRNGs because in some situations this property may be important.

 
 

-\section{Conclusion}

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