\usepackage{moreverb}
\usepackage{commath}
\usepackage{algorithm2e}
+\usepackage{listings}
\usepackage[standard]{ntheorem}
% Pour mathds : les ensembles IR, IN, etc.
\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}
-\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}
-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}
-Interet des itérations chaotiques pour générer des nombre alea\\
-Interet de générer des nombres alea 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.
-% >>>>>>>>>>>>>>>>>>>>>> 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}
-$f$ 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}
-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}
-$f$ 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}
-\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}
-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}
-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}
-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}
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
-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
-$$
+\begin{equation}
\forall n\in \mathds{N}^{\ast }, \forall i\in
\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
\begin{array}{ll}
x_i^{n-1} & \text{ if }S^n\neq i \\
\left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.
\end{array}\right.
-$$
+\end{equation}
\end{definition}
In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
$\left(f(x^{k})\right)_{S^{n}}$, 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}%
-\end{equation*}%
+\end{equation}%
\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},
-\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 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.
-\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:
-\begin{equation*}
+\begin{equation}
d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
-\end{equation*}
+\end{equation}
\noindent where
-\begin{equation*}
+\begin{equation}
\left\{
\begin{array}{lll}
\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
\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}
-\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}
-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}
-$G_{f}$ must be 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}
-The chaotic property of $G_f$ has been firstly established for the vectorial Boolean negation \cite{guyeux10}. To obtain a characterization, we have 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
-$\mathds{B}^n$; 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 $x'$.
-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}
-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}
-
-
+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}
+\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
-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}
-\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)}
$x\leftarrow x^0$\;
-$k\leftarrow b + \textit{XORshift}(b+1)$\;
-\For{$i=0,\dots,k-1$}
+$k\leftarrow b + \textit{XORshift}(b)$\;
+\For{$i=0,\dots,k$}
{
$s\leftarrow{\textit{XORshift}(n)}$\;
$x\leftarrow{F_f(s,x)}$\;
\end{algorithm}
\begin{algorithm}[h!]
-\SetAlgoLined
\KwIn{the internal configuration $z$ (a 32-bit word)}
\KwOut{$y$ (a 32-bit word)}
$z\leftarrow{z\oplus{(z\ll13)}}$\;
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
-\textit{XORshift}$(k)$ PRNGs \cite{Marsaglia2003} that returns integers uniformly distributed
+\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that returns integers
+uniformly distributed
into $\llbracket 1 ; k \rrbracket$.
-\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
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}.
-\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.
+\subsection{Improving the Speed of the Former Generator}
-\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}
+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:
+\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{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{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}
-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.
+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 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 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.
-Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
+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, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
-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}
+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}%
-\subsection{A given system can always be claimed as chaotic}
+Another time, a shift function appears as a component of these general chaotic
+iterations.
-Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point. Then this function is chaotic (in a certain way):
+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
+$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
-\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}
+The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
+\end{proposition}
\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.
+ $d_e$ 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}
+Before being able to study the topological behavior of the general
+chaotic iterations, we must firstly establish that:
+\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}
-\subsection{A given system can always be claimed as non-chaotic}
-
-\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}
\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 $y \notin I_x \Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
+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}
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,
+\begin{theorem}
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.
+\end{theorem}
+Let us firstly prove the following lemma.
+\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}
-\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.
-
-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{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 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 $s = 0,S^0~ S^1~ S^2~ \hdots = \sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$
+ \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}
-\end{definition}
+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}
+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}
-$\varphi$ 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:
+\section{Efficient PRNG based on Chaotic Iterations}
+\label{sec:efficient prng}
-\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
+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.
+
+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 $S^i$.
$$
-\begin{array}{cccl}
-s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 \rrbracket^{\mathds{N}} \\
- & x & \longmapsto & (s^k)_{k \in \mathds{N}}
-\end{array}
+\begin{array}{|cc|cccccccccccccccc|}
+\hline
+x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+\hline
+S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+\hline
+x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+\hline
+
+\hline
+ \end{array}
$$
-\end{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}
-\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: $$g(x) = \displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + \sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}.$$
+\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}
+
+
+
+
+
+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}.
+
+\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.
+
+\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{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]\;
+}
-\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}
+\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
+\label{algo:gpu_kernel}
+\end{algorithm}
-Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most usual one being the Euclidian distance recalled bellow:
+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 !!!}
-\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}
+\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}
-\medskip
+\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{algorithm}
+
+\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\;}
+
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadId is concerned} {
+ retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory and x\;
+ 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]\;
+}
-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:
+\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.
+
+\section{A cryptographically secure prng for GPU}
+
+It is possible to build a cryptographically secure prng based on the previous
+algorithm (algorithm~\ref{algo:gpu_kernel2}). It simply consists in replacing
+the {\it xor-like} algorithm by another cryptographically secure prng. In
+practice, we suggest to use the BBS algorithm~\cite{BBS} which takes the form:
+$$x_{n+1}=x_n^2~ mod~ M$$ where M is the product of the prime numbers. Those
+prime numbers need to be congruent to 3 modulus 4. In practice, this PRNG is
+known to be slow and not efficient for the generation of random numbers. For
+current GPU cards, the modulus operation is the most time consuming
+operation. So in order to obtain quite reasonable performances, it is required
+to use only modulus on 32 bits integer numbers. Consequently $x_n^2$ need to be
+less than $2^{32}$ and the number $M$ need to be less than $2^{16}$. So in
+pratice we can choose prime number around 256 that are congruent to 3 modulus 4.
+With 32 bits numbers, only the 4 least significant bits of $x_n$ can be chosen
+(the maximum number of undistinguishing is less or equals to
+$log_2(log_2(x_n))$).
+\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 consuming and slows down the random number
+generation. Moreover, if you are interested by applications that consume 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{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{figure}[htbp]
\begin{center}
-$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k, \check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$.
+ \includegraphics[scale=.7]{curve_time_gpu.pdf}
\end{center}
-\end{definition}
+\caption{Number of random numbers generated per second}
+\label{fig:time_gpu}
+\end{figure}
-\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}
+In comparison, Listing~\ref{algo:seqCIprng} allows us to generate about
+138MSample/s with only one core of the Xeon E5530.
-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}
+%% \section{Cryptanalysis of the Proposed PRNG}
+%% Mettre ici la preuve de PCH
-\subsubsection{The semiconjugacy}
+%\section{The relativity of disorder}
+%\label{sec:de la relativité du désordre}
-It is now possible to define a topological semiconjugacy between $\mathcal{X}$ and an interval of $\mathds{R}$:
+%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}.
-\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}
+%\subsection{Impact of the topology's finenesse}
-\begin{proof}
-$\varphi$ has been constructed in order to be continuous and onto.
-\end{proof}
+%Let us firstly introduce the following notations.
-In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} \big[$.
+%\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{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 $(\mathcal{X}_\tau,f)$.
-\subsection{Study of the chaotic iterations described as a real function}
+%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.
+%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 $V$.
-\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}
+%Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood
+%of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
+%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}
+%\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):
-\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{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{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}
+%\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.
-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:
+%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}
-\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}
-\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}
+%\subsection{A given system can always be claimed as non-chaotic}
-\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{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}
+%\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 $f$.
-\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}
+%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}
-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}
+%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}
-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{corollary}
+%$D$ is finer than the Euclidian distance $\Delta$.
+%\end{corollary}
-The sequential characterization of the continuity concludes the demonstration.
-\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}
+%\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
+%\label{chpt:Chaos des itérations chaotiques sur R}
-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 $x$.
+%\subsubsection{Chaos according to Devaney}
-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}
+%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}
-The conclusion of these propositions is that the proposed metric is more precise than the Euclidian distance, that is:
+%This result can be formulated as follows.
-\begin{corollary}
-$D$ is finer than the Euclidian distance $\Delta$.
-\end{corollary}
+%\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}
-This corollary can be reformulated as follows:
+%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.
+%
-\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}
-\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
-\label{chpt:Chaos des itérations chaotiques sur R}
-\subsubsection{Chaos according to Devaney}
+\section{Security Analysis}
-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.
-
+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{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.
-\section{Efficient prng based on chaotic iterations}
+\begin{proposition}
+If $H$ is a secure cryptographic PRNG, then $X$ is a secure cryptographic
+PRNG too.
+\end{proposition}
-On parle du séquentiel avec des nombres 64 bits\\
+\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}
-Faire le lien avec le paragraphe précédent (je considère que la stratégie s'appelle $S^i$\\
+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 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 $S^i$ 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 $S^i$. In
-order to obtain the strategy we also use a classical PRNG.
+\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}
-\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$>>$32);\\
- x = x\textasciicircum (unsigned int)(t3$>>$32);\\
- x = x\textasciicircum (unsigned int)t2;\\
- x = x\textasciicircum (unsigned int)(t1$>>$32);\\
- x = x\textasciicircum (unsigned int)t3;\\
- return x;\\
-\}
-\end{minipage}
-}
-\end{center}
-\caption{sequential Chaotic Iteration PRNG}
-\label{algo:seqCIprng}
-\end{figure}
-In Figure~\ref{algo:seqCIprng} a sequential version of our chaotic iterations
-based PRNG is presented. This version uses three classical 64 bits PRNG: the
-\texttt{xorshift}, the \texttt{xor128} and the \texttt{xorwow}. These three
-PRNGs are presented in~\cite{Marsaglia2003}. As each PRNG used works with
-64-bits and as our PRNG works with 32 bits, the use of \texttt{(unsigned int)}
-selects the 32 least significant bits whereas \texttt{(unsigned int)(t3$>>$32)}
-selects the 32 most significants bits of the variable \texttt{t}. This version
-sucesses the BigCrush of the TestU01 battery [P. L’ecuyer and
- R. Simard. Testu01].
-\section{Efficient prng based on chaotic iterations on GPU}
-On parle du passage du sequentiel au GPU
-\section{Experiments}
+\section{Conclusion}
-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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