\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 and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers 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, Christophe
+Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}
+
\maketitle
\begin{abstract}
-This is the abstract
+In this paper we present a new pseudorandom number generator (PRNG) on
+graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations. It
+is firstly proven to be chaotic according to the Devaney's formulation. We thus propose an efficient
+implementation for GPU that successfully passes the {\it BigCrush} tests, deemed to be the hardest
+battery of tests in TestU01. Experiments show that this PRNG can generate
+about 20 billions of random numbers per second on Tesla C1060 and NVidia GTX280
+cards.
+It is finally established that, under reasonable assumptions, the proposed PRNG can be cryptographically
+secure.
+
+
\end{abstract}
\section{Introduction}
-Interet des itérations chaotiques pour générer des nombre alea\\
-Interet de générer des nombres alea sur GPU
-\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
-...
-
+Randomness is of importance in many fields as scientific simulations or cryptography.
+``Random numbers'' can mainly be generated either by a deterministic and reproducible algorithm
+called a pseudorandom number generator (PRNG), or by a physical non-deterministic
+process having all the characteristics of a random noise, called a truly random number
+generator (TRNG).
+In this paper, we focus on reproducible generators, useful for instance in
+Monte-Carlo based simulators or in several cryptographic schemes.
+These domains need PRNGs that are statistically irreproachable.
+On some fields as in numerical simulations, speed is a strong requirement
+that is usually attained by using parallel architectures. In that case,
+a recurrent problem is that a deflate of the statistical qualities is often
+reported, when the parallelization of a good PRNG is realized.
+This is why ad-hoc PRNGs for each possible architecture must be found to
+achieve both speed and randomness.
+On the other side, speed is not the main requirement in cryptography: the great
+need is to define \emph{secure} generators being able to withstand malicious
+attacks. Roughly speaking, an attacker should not be able in practice to make
+the distinction between numbers obtained with the secure generator and a true random
+sequence.
+Finally, a small part of the community working in this domain focus on a
+third requirement, that is to define chaotic generators.
+The main idea is to take benefits from a chaotic dynamical system to obtain a
+generator that is unpredictable, disordered, sensible to its seed, or in other words chaotic.
+Their desire is to map a given chaotic dynamics into a sequence that seems random
+and unassailable due to chaos.
+However, the chaotic maps used as a pattern are defined in the real line
+whereas computers deal with finite precision numbers.
+This distortion leads to a deflation of both chaotic properties and speed.
+Furthermore, authors of such chaotic generators often claim their PRNG
+as secure due to their chaos properties, but there is no obvious relation
+between chaos and security as it is understood in cryptography.
+This is why the use of chaos for PRNG still remains marginal and disputable.
+
+The authors' opinion is that topological properties of disorder, as they are
+properly defined in the mathematical theory of chaos, can reinforce the quality
+of a PRNG. But they are not substitutable for security or statistical perfection.
+Indeed, to the authors' point of view, such properties can be useful in the two following situations. On the
+one hand, a post-treatment based on a chaotic dynamical system can be applied
+to a PRNG statistically deflective, in order to improve its statistical
+properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.
+On the other hand, chaos can be added to a fast, statistically perfect PRNG and/or a
+cryptographically secure one, in case where chaos can be of interest,
+\emph{only if these last properties are not lost during
+the proposed post-treatment}. Such an assumption is behind this research work.
+It leads to the attempts to define a
+family of PRNGs that are chaotic while being fast and statistically perfect,
+or cryptographically secure.
+Let us finish this paragraph by noticing that, in this paper,
+statistical perfection refers to the ability to pass the whole
+{\it BigCrush} battery of tests, which is widely considered as the most
+stringent statistical evaluation of a sequence claimed as random.
+This battery can be found into the well-known TestU01 package~\cite{LEcuyerS07}.
+Chaos, for its part, refers to the well-established definition of a
+chaotic dynamical system proposed by Devaney~\cite{Devaney}.
+
+
+In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave
+as a chaotic dynamical system. Such a post-treatment leads to a new category of
+PRNGs. We have shown that proofs of Devaney's chaos can be established for this
+family, and that the sequence obtained after this post-treatment can pass the
+NIST~\cite{Nist10}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} batteries of tests, even if the inputted generators
+cannot.
+The proposition of this paper is to improve widely the speed of the formerly
+proposed generator, without any lack of chaos or statistical properties.
+In particular, a version of this PRNG on graphics processing units (GPU)
+is proposed.
+Although 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 pseudorandom
+numbers inside a GPU when a scientific application runs in it. This remark
+motivates our proposal of a chaotic and statistically perfect PRNG for GPU.
+Such device
+allows us to generated almost 20 billions of pseudorandom numbers per second.
+Last, but not least, we show that the proposed post-treatment preserves the
+cryptographical security of the inputted PRNG, when this last has such a
+property.
+
+The remainder of this paper is organized as follows. In Section~\ref{section:related
+ works} we review some GPU implementations of PRNGs. Section~\ref{section:BASIC
+ RECALLS} gives some basic recalls on the well-known Devaney's formulation of chaos,
+ and on an iteration process called ``chaotic
+iterations'' on which the post-treatment is based.
+Proofs of chaos are given in Section~\ref{sec:pseudorandom}.
+Section~\ref{sec:efficient prng} presents an efficient
+implementation of this chaotic PRNG on a CPU, whereas Section~\ref{sec:efficient prng
+ gpu} describes the GPU implementation.
+Such generators are experimented in
+Section~\ref{sec:experiments}.
+We show in Section~\ref{sec:security analysis} that, if the inputted
+generator is cryptographically secure, then it is the case too for the
+generator provided by the post-treatment.
+Such a proof leads to the proposition of a cryptographically secure and
+chaotic generator on GPU based on the famous Blum Blum Shum
+in Section~\ref{sec:CSGPU}.
+This research work ends by a conclusion section, in which the contribution is
+summarized and intended future work is presented.
+
+
+
+
+\section{Related works on GPU based PRNGs}
+\label{section:related works}
+
+Numerous research works on defining GPU based PRNGs have yet been proposed in the
+literature, so that completeness is impossible.
+This is why authors of this document only give reference to the most significant attempts
+in this domain, from their subjective point of view.
+The quantity of pseudorandom numbers generated per second is mentioned here
+only when the information is given in the related work.
+A million numbers per second will be simply written as
+1MSample/s whereas a billion numbers per second is 1GSample/s.
+
+In \cite{Pang:2008:cec} a PRNG based on cellular automata is defined
+with no requirement to an high precision integer arithmetic or to any bitwise
+operations. Authors can generate about
+3.2MSamples/s on a GeForce 7800 GTX GPU, which is quite an old card now.
+However, there is neither a mention of statistical tests nor any proof of
+chaos or cryptography in this document.
+
+In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs
+based on Lagged Fibonacci or Hybrid Taus. They have used these
+PRNGs for Langevin simulations of biomolecules fully implemented on
+GPU. Performance of the GPU versions are far better than those obtained with a
+CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01.
+However the evaluations of the proposed PRNGs are only statistical ones.
+
+
+Authors of~\cite{conf/fpga/ThomasHL09} have studied the implementation of some
+PRNGs on different computing architectures: CPU, field-programmable gate array
+(FPGA), massively parallel processors, and GPU. This study is of interest, because
+the performance of the same PRNGs on different architectures are compared.
+FPGA appears as the fastest and the most
+efficient architecture, providing the fastest number of generated pseudorandom numbers
+per joule.
+However, we can notice that authors can ``only'' generate between 11 and 16GSamples/s
+with a GTX 280 GPU, which should be compared with
+the results presented in this document.
+We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only
+able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one.
+
+Lastly, Cuda has developed a library for the generation of pseudorandom numbers called
+Curand~\cite{curand11}. Several PRNGs are implemented, among
+other things
+Xorwow~\cite{Marsaglia2003} and some variants of Sobol. The tests reported show that
+their fastest version provides 15GSamples/s on the new Fermi C2050 card.
+But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
+\newline
+\newline
+We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded.
\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}
-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: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
+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 $V$. $f^{k}=f\circ ...\circ f$
+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}
-Let $f$ be a map from $\mathds{B}^n$ to itself. Then $G_{f}$ is continuous in the metric space $(\mathcal{X},d)$.
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. Then $G_{f}$ is continuous in
+the metric space $(\mathcal{X},d)$.
\end{proposition}
-The chaotic property of $G_f$ has been firstly established for the vectorial Boolean negation \cite{guyeux10}. To obtain a characterization, we have secondly introduced the notion of asynchronous iteration graph recalled bellow.
+The chaotic property of $G_f$ has been firstly established for the vectorial
+Boolean negation $f(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
+introduced the notion of asynchronous iteration graph recalled bellow.
-Let $f$ be a map from $\mathds{B}^n$ to itself. The
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
{\emph{asynchronous iteration graph}} associated with $f$ is the
directed graph $\Gamma(f)$ defined by: the set of vertices is
-$\mathds{B}^n$; 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 $\llbracket 1 ; n \rrbracket^{\mathds{N}} \times \mathds{B}^n$, is build from Boolean networks $f : \mathds{B}^n \rightarrow \mathds{B}^n$, we can preserve the theoretical properties on $G_f$ during implementations (due to the discrete nature of $f$). It is as if $\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ; n \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
+This result of chaos has lead us to study the possibility to build a
+pseudorandom number generator (PRNG) based on the chaotic iterations.
+As $G_f$, defined on the domain $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}}
+\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
+\rightarrow \mathds{B}^\mathsf{N}$, 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, in PRNG, or a physical noise in TRNG).
-\section{Application to Pseudo-Randomness}
+\section{Application to pseudorandomness}
+\label{sec:pseudorandom}
+
+\subsection{A First pseudorandom 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 %%RAPH: cette ligne provoque une erreur chez moi
\KwIn{the internal configuration $z$ (a 32-bit word)}
\KwOut{$y$ (a 32-bit word)}
$z\leftarrow{z\oplus{(z\ll13)}}$\;
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 $x^0$.
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques};
+an integer $b$, ensuring that the number of executed iterations is at least $b$
+and at most $2b+1$; and an initial configuration $x^0$.
It returns the new generated configuration $x$. Internally, it embeds two
-\textit{XORshift}$(k)$ PRNGs \cite{Marsaglia2003} that returns integers uniformly distributed
+\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~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
+\subsection{Improving the Speed of the Former Generator}
-\alert{Mettre encore un peu de blabla sur le PRNG, puis enchaîner en disant que, ok, on peut préserver le chaos quand on passe sur machine, mais que le chaos dont il s'agit a été prouvé pour une distance bizarroïde sur un espace non moins hémoroïde, d'où ce qui suit}
-
-
-
-\section{The relativity of disorder}
-\label{sec:de la relativité du désordre}
+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:
-\subsection{Impact of the topology's finenesse}
+\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:
-Let us firstly introduce the following notations.
+\begin{equation}
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+\begin{array}{ll}
+ x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+\end{array}\right.
+\label{eq:generalIC}
+\end{equation}
+where $f$ is the vectorial negation and $\forall n \in \mathds{N}$,
+$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
+$k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary
+decomposition of $S^n$ is 1. Such chaotic iterations are more general
+than the ones presented in Definition \ref{Def:chaotic iterations} for
+the fact that, instead of updating only one term at each iteration,
+we select a subset of components to change.
+
+
+Obviously, replacing Algorithm~\ref{CI Algorithm} by
+Equation~\ref{equation Oplus}, possible when the iteration function is
+the vectorial negation, leads to a speed improvement. However, proofs
+of chaos obtained in~\cite{bg10:ij} have been established
+only for chaotic iterations of the form presented in Definition
+\ref{Def:chaotic iterations}. The question is now to determine whether the
+use of more general chaotic iterations to generate pseudorandom numbers
+faster, does not deflate their topological chaos properties.
+
+\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+\label{deuxième def}
+Let us consider the discrete dynamical systems in chaotic iterations having
+the general form:
-\begin{notation}
-$\mathcal{X}_\tau$ will denote the topological space $\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply $\mathcal{V} (x)$, if there is no ambiguity).
-\end{notation}
+\begin{equation}
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+\begin{array}{ll}
+ x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+\end{array}\right.
+\label{general CIs}
+\end{equation}
+In other words, at the $n^{th}$ iteration, only the cells whose id is
+contained into the set $S^{n}$ are iterated.
+Let us now rewrite these general chaotic iterations as usual discrete dynamical
+system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
+is required in order to study the topological behavior of the system.
-\begin{theorem}
-\label{Th:chaos et finesse}
-Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t. $\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous both for $\tau$ and $\tau'$.
+Let us introduce the following function:
+\begin{equation}
+\begin{array}{cccc}
+ \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
+ & (i,X) & \longmapsto & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X, \end{array}\right.
+\end{array}
+\end{equation}
+where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
-If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then $(\mathcal{X}_\tau,f)$ is chaotic too.
-\end{theorem}
+Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
+\begin{equation}
+\begin{array}{lrll}
+F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
+\longrightarrow & \mathds{B}^{\mathsf{N}} \\
+& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi
+(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+\end{array}%
+\end{equation}%
+where + and . are the Boolean addition and product operations, and $\overline{x}$
+is the negation of the Boolean $x$.
+Consider the phase space:
+\begin{equation}
+\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
+\mathds{B}^\mathsf{N},
+\end{equation}
+\noindent and the map defined on $\mathcal{X}$:
+\begin{equation}
+G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
+\end{equation}
+\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
+(S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
+\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function}
+$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$.
+Then the general chaotic iterations defined in Equation \ref{general CIs} can
+be described by the following discrete dynamical system:
+\begin{equation}
+\left\{
+\begin{array}{l}
+X^0 \in \mathcal{X} \\
+X^{k+1}=G_{f}(X^k).%
+\end{array}%
+\right.
+\end{equation}%
-\begin{proof}
-Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
+Another time, a shift function appears as a component of these general chaotic
+iterations.
-Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in \tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) = \varnothing$. Consequently, $f$ is $\tau-$transitive.
+To study the Devaney's chaos property, a distance between two points
+$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
+Let us introduce:
+\begin{equation}
+d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
+\label{nouveau d}
+\end{equation}
+\noindent where
+\begin{equation}
+\left\{
+\begin{array}{lll}
+\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
+}\delta (E_{k},\check{E}_{k})}\textrm{ is another time the Hamming distance}, \\
+\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
+\sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
+\end{array}%
+\right.
+\end{equation}
+where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
+$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
-Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a periodic point for $f$ into $V$.
-Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
+\begin{proposition}
+The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
+\end{proposition}
-But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in \mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is proven.
+\begin{proof}
+ $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
+too, thus $d$ will be a distance as sum of two distances.
+ \begin{itemize}
+\item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then
+$d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then
+$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
+ \item $d_s$ is symmetric
+($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
+of the symmetric difference.
+\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$,
+and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$,
+we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
+inequality is obtained.
+ \end{itemize}
\end{proof}
-\subsection{A given system can always be claimed as chaotic}
-Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point. Then this function is chaotic (in a certain way):
+Before being able to study the topological behavior of the general
+chaotic iterations, we must firstly establish that:
-\begin{theorem}
-Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having at least a fixed point.
-Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete) topology on $\X$.
-\end{theorem}
+\begin{proposition}
+ For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on
+$\left( \mathcal{X},d\right)$.
+\end{proposition}
\begin{proof}
-$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus \{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq \varnothing$.
-As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For instance, $n=0$ is appropriate.
-
-Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V = \mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is regular, and the result is established.
+We use the sequential continuity.
+Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
+\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
+G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
+G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
+thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
+sequences).\newline
+As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
+to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
+d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
+In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
+cell will change its state:
+$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
+
+In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
+\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
+n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
+first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
+
+Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
+identical and strategies $S^n$ and $S$ start with the same first term.\newline
+Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
+so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
+\noindent We now prove that the distance between $\left(
+G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
+0. Let $\varepsilon >0$. \medskip
+\begin{itemize}
+\item If $\varepsilon \geqslant 1$, we see that distance
+between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
+strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
+\medskip
+\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
+\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
+\begin{equation*}
+\exists n_{2}\in \mathds{N},\forall n\geqslant
+n_{2},d_{s}(S^n,S)<10^{-(k+2)},
+\end{equation*}%
+thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
+\end{itemize}
+\noindent As a consequence, the $k+1$ first entries of the strategies of $%
+G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
+the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
+10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline
+In conclusion,
+$$
+\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
+,\forall n\geqslant N_{0},
+ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+\leqslant \varepsilon .
+$$
+$G_{f}$ is consequently continuous.
\end{proof}
-
-
-\subsection{A given system can always be claimed as non-chaotic}
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,
\begin{theorem}
-Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
-If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic (for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.
\end{theorem}
-\begin{proof}
-Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty}, f\right)$ is both transitive and regular.
-
-Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty}, f\right)$ is regular. Then $x$ must be a periodic point of $f$.
+Let us firstly prove the following lemma.
-Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite because $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in \mathcal{X}, y \notin I_x$.
+\begin{lemma}[Strong transitivity]
+\label{strongTrans}
+ For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can
+find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+\end{lemma}
-As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq \varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x \Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
+\begin{proof}
+ Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$.
+Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$,
+are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define
+$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+the form $(S',E')$ where $E'=E$ and $S'$ starts with
+$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+\begin{itemize}
+ \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+ \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+\end{itemize}
+Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$,
+where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+claimed in the lemma.
\end{proof}
+We can now prove the Theorem~\ref{t:chaos des general}...
+
+\begin{proof}[Theorem~\ref{t:chaos des general}]
+Firstly, strong transitivity implies transitivity.
+
+Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+prove that $G_f$ is regular, it is sufficient to prove that
+there exists a strategy $\tilde S$ such that the distance between
+$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+$(\tilde S,E)$ is a periodic point.
+
+Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$
+and $t_2\in\mathds{N}$ such
+that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+of $S$ and the first $t_2$ terms of $S'$: $$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
+$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
+point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
+have $d((S,E),(\tilde S,E))<\epsilon$.
+\end{proof}
+\section{Efficient PRNG based on Chaotic Iterations}
+\label{sec:efficient prng}
+In order to implement efficiently a PRNG based on chaotic iterations it is
+possible to improve previous works [ref]. One solution consists in considering
+that the strategy used contains all the bits for which the negation is
+achieved out. Then in order to apply the negation on these bits we can simply
+apply the xor operator between the current number and the strategy. In
+order to obtain the strategy we also use a classical PRNG.
-\section{Chaos on the order topology}
-
-\subsection{The phase space is an interval of the real line}
+Here is an example with 16-bits numbers showing how the bitwise operations
+are
+applied. Suppose that $x$ and the strategy $S^i$ are defined in binary mode.
+Then the following table shows the result of $x$ xor $S^i$.
+$$
+\begin{array}{|cc|cccccccccccccccc|}
+\hline
+x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+\hline
+S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+\hline
+x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+\hline
-\subsubsection{Toward a topological semiconjugacy}
+\hline
+ \end{array}
+$$
-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{itemize}
-\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$.
-\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots = \sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$
-\end{itemize}
-\end{definition}
+\lstset{language=C,caption={C code of the sequential chaotic iterations based
+PRNG},label=algo:seqCIprng}
+\begin{lstlisting}
+unsigned int CIprng() {
+ static unsigned int x = 123123123;
+ unsigned long t1 = xorshift();
+ unsigned long t2 = xor128();
+ unsigned long t3 = xorwow();
+ x = x^(unsigned int)t1;
+ x = x^(unsigned int)(t2>>32);
+ x = x^(unsigned int)(t3>>32);
+ x = x^(unsigned int)t2;
+ x = x^(unsigned int)(t1>>32);
+ x = x^(unsigned int)t3;
+ return x;
+}
+\end{lstlisting}
-$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic iterations $\Go$ on this real interval. To do so, two intermediate functions over $\big[ 0, 2^{10} \big[$ must be introduced:
-\begin{definition}
-\label{def:e et s}
-Let $x \in \big[ 0, 2^{10} \big[$ and:
-\begin{itemize}
-\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$: $\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$.
-\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal decomposition of $x$ is the one that does not have an infinite number of 9:
-$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$.
-\end{itemize}
-$e$ and $s$ are thus defined as follows:
-$$
-\begin{array}{cccl}
-e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\
- & x & \longmapsto & (e_0, \hdots, e_9)
-\end{array}
-$$
-\noindent and
-$$
-\begin{array}{cccl}
-s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 \rrbracket^{\mathds{N}} \\
- & x & \longmapsto & (s^k)_{k \in \mathds{N}}
-\end{array}
-$$
-\end{definition}
-We are now able to define the function $g$, whose goal is to translate the chaotic iterations $\Go$ on an interval of $\mathds{R}$.
+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$>>$32)} selects the 32 most significants bits of the
+variable \texttt{t}. So to produce a random number realizes 6 xor operations
+with 6 32-bits numbers produced by 3 64-bits PRNG. This version successes the
+BigCrush of the TestU01 battery~\cite{LEcuyerS07}.
-\begin{definition}
-$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}
+\section{Efficient PRNGs based on chaotic iterations on GPU}
+\label{sec:efficient prng gpu}
-\bigskip
+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}
-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: $$g(x) = \displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + \sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}.$$
+\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.
-\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}
+All the tests performed to pass the BigCrush of TestU01 succeeded. Different
+number of threads, called \texttt{NumThreads} in our algorithm, have been tested
+upto $10$ millions.
+\newline
+\newline
+{\bf QUESTION : on laisse cette remarque, je suis mitigé !!!}
-\medskip
+\begin{remark}
+Algorithm~\ref{algo:gpu_kernel} has the advantage to manipulate independent
+PRNGs, so this version is easily usable on a cluster of computer. The only thing
+to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in
+using a master node for the initialization which computes the initial parameters
+for all the differents nodes involves in the computation.
+\end{remark}
-This Euclidian distance does not reproduce exactly the notion of proximity induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$. This is the reason why we have to introduce the following metric:
+\subsection{Improved version for GPU}
+As GPU cards using CUDA have shared memory between threads of the same block, it
+is possible to use this feature in order to simplify the previous algorithm,
+i.e., using less than 3 xor-like PRNGs. The solution consists in computing only
+one xor-like PRNG by thread, saving it into shared memory and using the results
+of some other threads in the same block of threads. In order to define which
+thread uses the result of which other one, we can use a permutation array which
+contains the indexes of all threads and for which a permutation has been
+performed. In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used.
+The variable \texttt{offset} is computed using the value of
+\texttt{permutation\_size}. Then we can compute \texttt{o1} and \texttt{o2}
+which represent the indexes of the other threads for which the results are used
+by the current thread. In the algorithm, we consider that a 64-bits xor-like
+PRNG is used, that is why both 32-bits parts are used.
+This version also succeeds to the {\it BigCrush} batteries of tests.
-\begin{definition}
-Let $x,y \in \big[ 0, 2^{10} \big[$.
-$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$ defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$, where:
-\begin{center}
-$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k, \check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$.
-\end{center}
-\end{definition}
+\begin{algorithm}
-\begin{proposition}
-$D$ is a distance on $\big[ 0, 2^{10} \big[$.
-\end{proposition}
+\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
+in global memory\;
+NumThreads: Number of threads\;
+tab1, tab2: Arrays containing permutations of size permutation\_size\;}
-\begin{proof}
-The three axioms defining a distance must be checked.
-\begin{itemize}
-\item $D \geqslant 0$, because everything is positive in its definition. If $D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal (they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then $\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$.
-\item $D(x,y)=D(y,x)$.
-\item Finally, the triangular inequality is obtained due to the fact that both $\delta$ and $\Delta(x,y)=|x-y|$ satisfy it.
-\end{itemize}
-\end{proof}
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadId is concerned} {
+ retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory 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]\;
+}
-The convergence of sequences according to $D$ is not the same than the usual convergence related to the Euclidian metric. For instance, if $x^n \to x$ according to $D$, then necessarily the integral part of each $x^n$ is equal to the integral part of $x$ (at least after a given threshold), and the decimal part of $x^n$ corresponds to the one of $x$ ``as far as required''.
-To illustrate this fact, a comparison between $D$ and the Euclidian distance is given Figure \ref{fig:comparaison de distances}. These illustrations show that $D$ is richer and more refined than the Euclidian distance, and thus is more precise.
+\caption{main kernel for the chaotic iterations based PRNG GPU efficient
+version}
+\label{algo:gpu_kernel2}
+\end{algorithm}
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in three operations having
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, three iterations of the general chaotic
+iterations are realized between two stored values of the PRNG.
+To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+we must guarantee that this dynamical system iterates on the space
+$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
+The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+To prevent from any flaws of chaotic properties, we must check that each right
+term, corresponding to terms of the strategies, can possibly be equal to any
+integer of $\llbracket 1, \mathsf{N} \rrbracket$.
+
+Such a result is obvious for the two first lines, as for the xor-like(), all the
+integers belonging into its interval of definition can occur at each iteration.
+It can be easily stated for the two last lines by an immediate mathematical
+induction.
+
+Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
+chaotic iterations presented previously, and for this reason, it satisfies the
+Devaney's formulation of a chaotic behavior.
-\begin{figure}[t]
+\section{Experiments}
+\label{sec:experiments}
+
+Different experiments have been performed in order to measure the generation
+speed. We have used a computer equiped with Tesla C1060 NVidia GPU card and an
+Intel Xeon E5530 cadenced at 2.40 GHz for our experiments and we have used
+another one equipped with a less performant CPU and a GeForce GTX 280. Both
+cards have 240 cores.
+
+In Figure~\ref{fig:time_xorlike_gpu} we compare the number of random numbers
+generated per second with the xor-like based PRNG. In this figure, the optimized
+version use the {\it xor64} described in~\cite{Marsaglia2003}. The naive version
+use the three xor-like PRNGs described in Listing~\ref{algo:seqCIprng}. In
+order to obtain the optimal performance we removed the storage of random numbers
+in the GPU memory. This step is time consuming and slows down the random numbers
+generation. Moreover, if one is interested by applications that consume random
+numbers directly when they are generated, their storage are completely
+useless. In this figure we can see that when the number of threads is greater
+than approximately 30,000 upto 5 millions the number of random numbers generated
+per second is almost constant. With the naive version, it is between 2.5 and
+3GSample/s. With the optimized version, it is approximately equals to
+20GSample/s. Finally we can remark that both GPU cards are quite similar. In
+practice, the Tesla C1060 has more memory than the GTX 280 and this memory
+should be of better quality.
+
+\begin{figure}[htbp]
\begin{center}
- \subfigure[Function $x \to dist(x;1,234) $ on the interval $(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad
- \subfigure[Function $x \to dist(x;3) $ on the interval $(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}}
+ \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf}
\end{center}
-\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).}
-\label{fig:comparaison de distances}
+\caption{Number of random numbers generated per second with the xorlike based PRNG}
+\label{fig:time_xorlike_gpu}
\end{figure}
+In comparison, Listing~\ref{algo:seqCIprng} allows us to generate about
+138MSample/s with only one core of the Xeon E5530.
-\subsubsection{The semiconjugacy}
+In Figure~\ref{fig:time_bbs_gpu} we highlight the performance of the optimized
+BBS based PRNG on GPU. Performances are less important. On the Tesla C1060 we
+obtain approximately 1.8GSample/s and on the GTX 280 about 1.6GSample/s.
-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}
+\begin{figure}[htbp]
+\begin{center}
+ \includegraphics[scale=.7]{curve_time_bbs_gpu.pdf}
+\end{center}
+\caption{Number of random numbers generated per second with the BBS based PRNG}
+\label{fig:time_bbs_gpu}
+\end{figure}
-In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} \big[$.
+Both these experimentations allows us to conclude that it is possible to
+generate a huge number of pseudorandom numbers with the xor-like version and
+about tens times less with the BBS based version. The former version has only
+chaotic properties whereas the latter also has cryptographically properties.
+%% \section{Cryptanalysis of the Proposed PRNG}
+%% Mettre ici la preuve de PCH
+%\section{The relativity of disorder}
+%\label{sec:de la relativité du désordre}
-\subsection{Study of the chaotic iterations described as a real function}
+%In the next two sections, we investigate the impact of the choices that have
+%lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.
+%\subsection{Impact of the topology's finenesse}
-\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 us firstly introduce the following notations.
+%\begin{notation}
+%$\mathcal{X}_\tau$ will denote the topological space
+%$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
+%of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
+%$\mathcal{V} (x)$, if there is no ambiguity).
+%\end{notation}
-\begin{figure}[t]
-\begin{center}
- \subfigure[ICs on the interval $(510;514)$.]{\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}
+%\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 $\tau'$.
-\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}
+%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 $(\mathcal{X}_\tau,f)$.
-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 $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in
+%\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we
+%can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) =
+%\varnothing$. Consequently, $f$ is $\tau-$transitive.
-\begin{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 $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
+%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 $V$.
-Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, $g$ is a linear function, having a slope equal to 10: $\forall x \notin I, g'(x)=10$.
-\end{proposition}
+%Let $x \in \mathcal{X}$ 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}
-\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 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}
+%Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
+%Then this function is chaotic (in a certain way):
+%\begin{theorem}
+%Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having
+%at least a fixed point.
+%Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete)
+%topology on $\X$.
+%\end{theorem}
-\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}
+%\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.
-The two propositions bellow allow to compare our two distances on $\big[ 0, 2^\mathsf{N} \big[$:
+%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}
-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}
+%\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}
-A contrario:
+%\begin{proof}
+%Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
+%f\right)$ is both transitive and regular.
-\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}
+%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 $f$.
-\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 $x$.
+%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}
-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}
+%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{corollary}
+%$D$ is finer than the Euclidian distance $\Delta$.
+%\end{corollary}
+
+%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}
-\begin{corollary}
-$D$ is finer than the Euclidian distance $\Delta$.
-\end{corollary}
-This corollary can be reformulated as follows:
+%\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
+%\label{chpt:Chaos des itérations chaotiques sur R}
-\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}
+%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.
-\subsubsection{Chaos according to Devaney}
+%\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}
-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}
+%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.
+%
-This result can be formulated as follows.
-\begin{theorem}
-\label{th:IC et topologie de l'ordre}
-The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the order topology.
-\end{theorem}
-Indeed this result is weaker than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
-
-\section{Efficient prng based on chaotic iterations}
+\section{Security Analysis}
+\label{sec:security analysis}
-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 bit operations are
-applied. Suppose that $x$ and the strategy $S^i$ are defined in binary mode.
-Then the following table shows the result of $x$ xor $S^i$.
-$$
-\begin{array}{|cc|cccccccccccccccc|}
-\hline
-x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
-\hline
-S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
-\hline
-x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
-\hline
-\hline
- \end{array}
-$$
+In this section the concatenation of two strings $u$ and $v$ is classically
+denoted by $uv$.
+In a cryptographic context, a pseudorandom generator is a deterministic
+algorithm $G$ transforming strings into strings and such that, for any
+seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
+$\ell_G(N)$ with $\ell_G(N)>N$.
+The notion of {\it secure} PRNGs can now be defined as follows.
-%% \begin{figure}[htbp]
-%% \begin{center}
-%% \fbox{
-%% \begin{minipage}{14cm}
-%% unsigned int CIprng() \{\\
-%% static unsigned int x = 123123123;\\
-%% unsigned long t1 = xorshift();\\
-%% unsigned long t2 = xor128();\\
-%% unsigned long t3 = xorwow();\\
-%% x = x\textasciicircum (unsigned int)t1;\\
-%% x = x\textasciicircum (unsigned int)(t2$>>$32);\\
-%% x = x\textasciicircum (unsigned int)(t3$>>$32);\\
-%% x = x\textasciicircum (unsigned int)t2;\\
-%% x = x\textasciicircum (unsigned int)(t1$>>$32);\\
-%% x = x\textasciicircum (unsigned int)t3;\\
-%% return x;\\
-%% \}
-%% \end{minipage}
-%% }
-%% \end{center}
-%% \caption{sequential Chaotic Iteration PRNG}
-%% \label{algo:seqCIprng}
-%% \end{figure}
-
-
-
-\lstset{language=C,caption={C code of the sequential chaotic iterations based PRNG},label=algo:seqCIprng}
-\begin{lstlisting}
-unsigned int CIprng() {
- static unsigned int x = 123123123;
- unsigned long t1 = xorshift();
- unsigned long t2 = xor128();
- unsigned long t3 = xorwow();
- x = x^(unsigned int)t1;
- x = x^(unsigned int)(t2>>32);
- x = x^(unsigned int)(t3>>32);
- x = x^(unsigned int)t2;
- x = x^(unsigned int)(t1>>32);
- x = x^(unsigned int)t3;
- return x;
-}
-\end{lstlisting}
+\begin{definition}
+A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
+algorithm $D$, for any positive polynomial $p$, and for all sufficiently
+large $k$'s,
+$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)}=1]|< \frac{1}{p(N)},$$
+where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the
+probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the
+internal coin tosses of $D$.
+\end{definition}
+Intuitively, it means that there is no polynomial time algorithm that can
+distinguish a perfect uniform random generator from $G$ with a non
+negligible probability. The interested reader is referred
+to~\cite[chapter~3]{Goldreich} for more information. Note that it is
+quite easily possible to change the function $\ell$ into any polynomial
+function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}.
+
+The generation schema developed in (\ref{equation Oplus}) is based on a
+pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,
+without loss of generality, that for any string $S_0$ of size $N$, the size
+of $H(S_0)$ is $kN$, with $k>2$. It means that $\ell_H(N)=kN$.
+Let $S_1,\ldots,S_k$ be the
+strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concatenation of
+the $S_i$'s). The cryptographic PRNG $X$ defined in (\ref{equation Oplus})
+is the algorithm mapping any string of length $2N$ $x_0S_0$ into the string
+$(x_0\oplus S_0 \oplus S_1)(x_0\oplus S_0 \oplus S_1\oplus S_2)\ldots
+(x_o\bigoplus_{i=0}^{i=k}S_i)$. Particularly one has $\ell_{X}(2N)=kN=\ell_H(N)$.
+We claim now that if this PRNG is secure,
+then the new one is secure too.
+\begin{proposition}
+If $H$ is a secure cryptographic PRNG, then $X$ is a secure cryptographic
+PRNG too.
+\end{proposition}
+\begin{proof}
+The proposition is proved by contraposition. Assume that $X$ is not
+secure. By Definition, there exists a polynomial time probabilistic
+algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
+$N\geq \frac{k_0}{2}$ satisfying
+$$| \mathrm{Pr}[D(X(U_{2N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)}.$$
+We describe a new probabilistic algorithm $D^\prime$ on an input $w$ of size
+$kN$:
+\begin{enumerate}
+\item Decompose $w$ into $w=w_1\ldots w_{k}$, where each $w_i$ has size $N$.
+\item Pick a string $y$ of size $N$ uniformly at random.
+\item Compute $z=(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y
+ \bigoplus_{i=1}^{i=k} w_i).$
+\item Return $D(z)$.
+\end{enumerate}
+
+
+Consider for each $y\in \mathbb{B}^{kN}$ the function $\varphi_{y}$
+from $\mathbb{B}^{kN}$ into $\mathbb{B}^{kN}$ mapping $w=w_1\ldots w_k$
+(each $w_i$ has length $N$) to
+$(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y
+ \bigoplus_{i=1}^{i=k_1} w_i).$ By construction, one has for every $w$,
+\begin{equation}\label{PCH-1}
+D^\prime(w)=D(\varphi_y(w)),
+\end{equation}
+where $y$ is randomly generated.
+Moreover, for each $y$, $\varphi_{y}$ is injective: if
+$(y\oplus w_1)(y\oplus w_1\oplus w_2)\ldots (y\bigoplus_{i=1}^{i=k_1}
+w_i)=(y\oplus w_1^\prime)(y\oplus w_1^\prime\oplus w_2^\prime)\ldots
+(y\bigoplus_{i=1}^{i=k} w_i^\prime)$, then for every $1\leq j\leq k$,
+$y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. It follows,
+by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
+is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
+one has
+\begin{equation}\label{PCH-2}
+\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1].
+\end{equation}
+Now, using (\ref{PCH-1}) again, one has for every $x$,
+\begin{equation}\label{PCH-3}
+D^\prime(H(x))=D(\varphi_y(H(x))),
+\end{equation}
+where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$,
+thus
+\begin{equation}\label{PCH-3}
+D^\prime(H(x))=D(yx),
+\end{equation}
+where $y$ is randomly generated.
+It follows that
-In listing~\ref{algo:seqCIprng} a sequential version of our chaotic iterations
-based PRNG is presented. 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 [P. L’ecuyer and
- R. Simard. Testu01].
+\begin{equation}\label{PCH-4}
+\mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1].
+\end{equation}
+ From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that
+there exist a polynomial time probabilistic
+algorithm $D^\prime$, a positive polynomial $p$, such that for all $k_0$ there exists
+$N\geq \frac{k_0}{2}$ satisfying
+$$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$
+proving that $H$ is not secure, a contradiction.
+\end{proof}
-\section{Efficient prng based on chaotic iterations on GPU}
-In order to benefit from computing power of GPU, a program needs to define
-independent blocks of threads which can be computed simultaneously. In general,
-the larger the number of threads is, the more local memory is used and the less
-branching instructions are used (if, while, ...), the better performance is
-obtained on GPU. So with algorithm \ref{algo:seqCIprng} presented in the
-previous section, it is possible to build a similar program which computes PRNG
-on GPU. The principe consists in assigning the computation of a PRNG as in
-sequential to each thread of the GPU. Of course, it is essential that the three
-xor-like PRNGs used for our computation have different parameters. So we chose
-them randomly with another PRNG. As the initialisation is performed by the CPU,
-we have chosen to use the ISAAC PRNG [ref] 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{threadId is concerned} {
- retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
- \For{i=1 to n} {
- compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
- store the new PRNG in NewNb[NumThreads*threadId+i]\;
- }
- store internal variables in InternalVarXorLikeArray[threadId]\;
-}
+\section{A cryptographically secure prng for GPU}
+\label{sec:CSGPU}
+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 two 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 numbers around 256 that are congruent to 3 modulus
+4. With 32 bits numbers, only the 4 least significant bits of $x_n$ can be
+chosen (the maximum number of undistinguishing is less or equals to
+$log_2(log_2(x_n))$). So to generate a 32 bits number, we need to use 8 times
+the BBS algorithm, with different combinations of $M$ is required.
-\caption{main kernel for the chaotic iterations based PRNG GPU version}
-\label{algo:gpu_kernel}
-\end{algorithm}
+Currently this PRNG does not succeed to pass all the tests of TestU01.
-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 have been tested upto $10$ millions.
+\section{Conclusion}
-\section{Experiments}
-Differents experiments have been performed in order to measure the generation speed.
+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.
+We also propose a PRNG cryptographically secure and its implementation on GPU.
+An efficient implementation on GPU based on a xor-like PRNG allows us to
+generate a huge number of pseudorandom numbers per second (about
+20Gsample/s). This PRNG succeeds to pass the hardest batteries of TestU01.
-First of all we have compared the time to generate X random numbers with both the CPU version and the GPU version.
+In future work we plan to extend this work for parallel PRNG for clusters or
+grid computing. We also plan to improve the BBS version in order to succeed all
+the tests of TestU01.
-Faire une courbe du nombre de random en fonction du nombre de threads, éventuellement en fonction du nombres de threads par bloc.
-\section{Conclusion}
-\bibliographystyle{plain}
+\bibliographystyle{plain}
\bibliography{mabase}
\end{document}
