\maketitle
\begin{abstract}
+In this paper we present a new produce pseudo-random numbers generator (PRNG) on
+graphics processing units (GPU). This PRNG is based on chaotic iterations. it
+is proven to be chaotic in the Devany's formulation. We propose an efficient
+implementation for GPU which succeeds to the {\it BigCrush}, the hardest
+batteries of test of TestU01. Experimentations show that this PRNG can generate
+about 20 billions of random numbers per second on Tesla C1060 and NVidia GTX280
+cards.
+
\end{abstract}
mathematical formulation of chaotic dynamical systems.
In a previous work~\cite{bgw09:ip} we have proposed a new familly of chaotic
-PRNG based on chaotic iterations (IC). We have proven that these PRNGs are
+PRNG based on chaotic iterations. We have proven that these PRNGs are
chaotic in the Devaney's sense. In this paper we propose a faster version which
is also proven to be chaotic.
the manipulation of images, they are nowadays commonly used in many scientific
applications. Therefore, it is important to be able to generate pseudo-random
numbers inside a GPU when a scientific application runs in a GPU. That is why we
-also provide an efficient PRNG for GPU respecting based on IC.
+also provide an efficient PRNG for GPU respecting based on IC. Such devices
+allows us to generated almost 20 billions of random numbers per second.
+In order to establish that our PRNGs are chaotic according to the Devaney's
+formulation, we extend what we have proposed in~\cite{guyeux10}. Moreover, we
+define a new distance to measure the disorder in the chaos and we prove some
+interesting properties with this distance.
+The rest of this paper is organised as follows. In Section~\ref{section:related
+ works} we review some GPU implementions of PRNG. Section~\ref{section:BASIC
+ RECALLS} gives some basic recalls on Devanay's formation of chaos and chaotic
+iterations. In Section~\ref{sec:pseudo-random} the proof of chaos of our PRNGs
+is studied. Section~\ref{sec:efficient prng} presents an efficient
+implementation of our chaotic PRNG on a CPU. Section~\ref{sec:efficient prng
+ gpu} describes the GPU implementation of our chaotic PRNG. In
+Section~\ref{sec:experiments} some experimentations are presented.
+Section~\ref{sec:de la relativité du désordre} describes the relativity of
+disorder. In Section~\ref{sec: chaos order topology} the proof that chaotic
+iterations can be described by iterations on a real interval is
+established. Finally, we give a conclusion and some perspectives.
-Interet des itérations chaotiques pour générer des nombre alea\\
-Interet de générer des nombres alea sur GPU
\section{Related works on GPU based PRNGs}
-
+\label{section:related works}
In the litterature many authors have work on defining GPU based PRNGs. We do not
want to be exhaustive and we just give the most significant works from our point
-of view.
+of view. When authors mention the number of random numbers generated per second
+we mention it. We consider that a million numbers per second corresponds to
+1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
In \cite{Pang:2008:cec}, the authors define a PRNG based on cellular automata
which does not require high precision integer arithmetics nor bitwise
operations. There is no mention of statistical tests nor proof that this PRNG is
-chaotic. Concerning the speed of generation, they can generate about 3200000
-random numbers per seconds on a GeForce 7800 GTX GPU (which is quite old now).
+chaotic. Concerning the speed of generation, they can generate about
+3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs
based on Lagged Fibonacci, Hybrid Taus or Hybrid Taus. They have used these
CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
no mention that their PRNGs have chaos mathematical properties.
+
+Authors of~\cite{conf/fpga/ThomasHL09} have studied the implementation of some
+PRNGs on diferrent computing architectures: CPU, field-programmable gate array
+(FPGA), GPU and massively parallel processor. This study is interesting because
+it shows the performance of the same PRNGs on different architeture. For
+example, the FPGA is globally the fastest architecture and it is also the
+efficient one because it provides the fastest number of generated random numbers
+per joule. Concerning the GPU, authors can generate betweend 11 and 16GSample/s
+with a GTX 280 GPU. The drawback of this work is that those PRNGs only succeed
+the {\it Crush} test which is easier than the {\it Big Crush} test.
+\newline
+\newline
To the best of our knowledge no GPU implementation have been proven to have chaotic properties.
\section{Basic Recalls}
\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
\section{Application to Pseudo-Randomness}
-
+\label{sec:pseudo-random}
\subsection{A First Pseudo-Random Number Generator}
We have proposed in~\cite{bgw09:ip} a new family of generators that receives
\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
\end{array}
$$
-%% \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}
+
with 6 32-bits numbers produced by 3 64-bits PRNG. This version successes the
BigCrush of the TestU01 battery~\cite{LEcuyerS07}.
-\section{Efficient prng based on chaotic iterations on GPU}
+\section{Efficient PRNGs based on chaotic iterations on GPU}
+\label{sec:efficient prng gpu}
In order to benefit from computing power of GPU, a program needs to define
independent blocks of threads which can be computed simultaneously. In general,
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 [ref] environment, threads have a local identificator,
-called \texttt{ThreadIdx} relative to the block containing them.
+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}
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.
+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}
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 succeed to the BigCrush batteries of tests.
+This version also succeeds to the {\it BigCrush} batteries of tests.
\begin{algorithm}
\KwOut{NewNb: array containing random numbers in global memory}
\If{threadId is concerned} {
- retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
+ retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory\;
offset = threadIdx\%permutation\_size\;
o1 = threadIdx-offset+tab1[offset]\;
o2 = threadIdx-offset+tab2[offset]\;
\For{i=1 to n} {
t=xor-like()\;
- shared\_mem[threadId]=(unsigned int)t\;
- x = x $\oplus$ (unsigned int) t\;
- x = x $\oplus$ (unsigned int) (t>>32)\;
- x = x $\oplus$ shared[o1]\;
- x = x $\oplus$ shared[o2]\;
+ 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]\;
}
\subsection{Theoretical Evaluation of the Improved Version}
-A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having
+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, four iterations of the general chaotic
+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
Devaney's formulation of a chaotic behavior.
\section{Experiments}
-
-Different experiments have been performed in order to measure the generation
-speed.
-\begin{figure}[t]
+\label{sec:experiments}
+
+Different experiments have been performed in order to measure the generation
+speed. We have used a computer equiped with Tesla C1060 NVidia GPU card and an
+Intel Xeon E5530 cadenced at 2.40 GHz for our experiments and we have used
+another one equipped with a less performant CPU and a GeForce GTX 280. Both
+cards have 240 cores.
+
+In Figure~\ref{fig:time_gpu} we compare the number of random numbers generated
+per second. The xor-like prng is a xor64 described in~\cite{Marsaglia2003}. In
+order to obtain the optimal performance we remove the storage of random numbers
+in the GPU memory. This step is time consumming and slows down the random number
+generation. Moreover, if you are interested by applications that consome random
+numbers directly when they are generated, their storage is completely
+useless. In this figure we can see that when the number of threads is greater
+than approximately 30,000 upto 5 millions the number of random numbers generated
+per second is almost constant. With the naive version, it is between 2.5 and
+3GSample/s. With the optimized version, it is approximately equals to
+20GSample/s. Finally we can remark that both GPU cards are quite similar. In
+practice, the Tesla C1060 has more memory than the GTX 280 and this memory
+should be of better quality.
+
+\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.7]{curve_time_gpu.pdf}
\end{center}
\caption{Number of random numbers generated per second}
-\label{fig:time_naive_gpu}
-\end{figure}
-
-
-First of all we have compared the time to generate X random numbers with both
-the CPU version and the GPU version.
-
-Faire une courbe du nombre de random en fonction du nombre de threads,
-éventuellement en fonction du nombres de threads par bloc.
-
-
-
-\section{The relativity of disorder}
-\label{sec:de la relativité du désordre}
-
-In the next two sections, we investigate the impact of the choices that have
-lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.
-
-\subsection{Impact of the topology's finenesse}
-
-Let us firstly introduce the following notations.
-
-\begin{notation}
-$\mathcal{X}_\tau$ will denote the topological space
-$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
-of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
-$\mathcal{V} (x)$, if there is no ambiguity).
-\end{notation}
-
-
-
-\begin{theorem}
-\label{Th:chaos et finesse}
-Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t.
-$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous
-both for $\tau$ and $\tau'$.
-
-If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then
-$(\mathcal{X}_\tau,f)$ is chaotic too.
-\end{theorem}
-
-\begin{proof}
-Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
-
-Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in
-\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we
-can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) =
-\varnothing$. Consequently, $f$ is $\tau-$transitive.
-
-Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for
-all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a
-periodic point for $f$ into $V$.
-
-Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood
-of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
-
-But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in
-\mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a
-periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is
-proven.
-\end{proof}
-
-\subsection{A given system can always be claimed as chaotic}
-
-Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
-Then this function is chaotic (in a certain way):
-
-\begin{theorem}
-Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having
-at least a fixed point.
-Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete)
-topology on $\X$.
-\end{theorem}
-
-
-\begin{proof}
-$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus
-\{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq
-\varnothing$.
-As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for
-an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For
-instance, $n=0$ is appropriate.
-
-Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V =
-\mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is
-regular, and the result is established.
-\end{proof}
-
-
-
-
-\subsection{A given system can always be claimed as non-chaotic}
-
-\begin{theorem}
-Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
-If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic
-(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
-\end{theorem}
-
-\begin{proof}
-Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
-f\right)$ is both transitive and regular.
-
-Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must
-contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty},
-f\right)$ is regular. Then $x$ must be a periodic point of $f$.
-
-Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite
-because $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in
-\mathcal{X}, y \notin I_x$.
-
-As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty
-sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq
-\varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x
-\Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
-\end{proof}
-
-
-
-
-
-
-\section{Chaos on the order topology}
-
-\subsection{The phase space is an interval of the real line}
-
-\subsubsection{Toward a topological semiconjugacy}
-
-In what follows, our intention is to establish, by using a topological
-semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as
-iterations on a real interval. To do so, we must firstly introduce some
-notations and terminologies.
-
-Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket
-1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N}
-\times \B^\mathsf{N}$.
-
-
-\begin{definition}
-The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[
-0, 2^{10} \big[$ is defined by:
-\begin{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}
+\label{fig:time_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}
-It is now possible to define a topological semiconjugacy between $\mathcal{X}$
-and an interval of $\mathds{R}$:
-\begin{theorem}
-Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on
-$\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow:
-\begin{equation*}
-\begin{CD}
-\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>>
-\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\
- @V{\varphi}VV @VV{\varphi}V\\
-\left( ~\big[ 0, 2^{10} \big[, D~\right) @>>g> \left(~\big[ 0, 2^{10} \big[,
-D~\right)
-\end{CD}
-\end{equation*}
-\end{theorem}
-\begin{proof}
-$\varphi$ has been constructed in order to be continuous and onto.
-\end{proof}
+\section{Cryptanalysis of the Proposed PRNG}
-In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N}
-\big[$.
+Mettre ici la preuve de PCH
+%\section{The relativity of disorder}
+%\label{sec:de la relativité du désordre}
+%In the next two sections, we investigate the impact of the choices that have
+%lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.
+%\subsection{Impact of the topology's finenesse}
+%Let us firstly introduce the following notations.
-\subsection{Study of the chaotic iterations described as a real function}
+%\begin{notation}
+%$\mathcal{X}_\tau$ will denote the topological space
+%$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
+%of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
+%$\mathcal{V} (x)$, if there is no ambiguity).
+%\end{notation}
-\begin{figure}[t]
-\begin{center}
- \subfigure[ICs on the interval
-$(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad
- \subfigure[ICs on the interval
-$(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\
- \subfigure[ICs on the interval
-$(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad
- \subfigure[ICs on the interval
-$(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}}
-\end{center}
-\caption{Representation of the chaotic iterations.}
-\label{fig:ICs}
-\end{figure}
-
-
-
-
-\begin{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}
-
-
-\begin{proof}
-Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket
-0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral
-prat $e$ and the same decimal part $s^0$: on the set $I_n$, functions $e(x)$
-and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all
-the images $g(x)$ of these points $x$:
-\begin{itemize}
-\item Have the same integral part, which is $e$, except probably the bit number
-$s^0$. In other words, this integer has approximately the same binary
-decomposition than $e$, the sole exception being the digit $s^0$ (this number is
-then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$,
-\emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$).
-\item A shift to the left has been applied to the decimal part $y$, losing by
-doing so the common first digit $s^0$. In other words, $y$ has been mapped into
-$10\times y - s^0$.
-\end{itemize}
-To sum up, the action of $g$ on the points of $I$ is as follows: first, make a
-multiplication by 10, and second, add the same constant to each term, which is
-$\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$.
-\end{proof}
-\begin{remark}
-Finally, chaotic iterations are elements of the large family of functions that
-are both chaotic and piecewise linear (like the tent map).
-\end{remark}
+%\begin{theorem}
+%\label{Th:chaos et finesse}
+%Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t.
+%$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous
+%both for $\tau$ and $\tau'$.
+%If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then
+%$(\mathcal{X}_\tau,f)$ is chaotic too.
+%\end{theorem}
+%\begin{proof}
+%Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
-\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}
+%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.
-The two propositions bellow allow to compare our two distances on $\big[ 0,
-2^\mathsf{N} \big[$:
+%Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for
+%all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a
+%periodic point for $f$ into $V$.
-\begin{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}
+%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{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}
+%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}
-The sequential characterization of the continuity concludes the demonstration.
-\end{proof}
+%\subsection{A given system can always be claimed as chaotic}
+%Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
+%Then this function is chaotic (in a certain way):
+%\begin{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}
-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}
+%$f$ 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.
-\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:
+%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{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}
+%\subsection{A given system can always be claimed as non-chaotic}
-\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
-\label{chpt:Chaos des itérations chaotiques sur R}
+%\begin{theorem}
+%Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
+%If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic
+%(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
+%\end{theorem}
+%\begin{proof}
+%Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
+%f\right)$ is both transitive and regular.
+%Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must
+%contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty},
+%f\right)$ is regular. Then $x$ must be a periodic point of $f$.
-\subsubsection{Chaos according to Devaney}
+%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}
-We have recalled previously that the chaotic iterations $\left(\Go,
-\mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We
-can deduce that they are chaotic on $\mathds{R}$ too, when considering the order
-topology, because:
-\begin{itemize}
-\item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10}
-\big[_D\right)$ are semiconjugate by $\varphi$,
-\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic
-according to Devaney, because the semiconjugacy preserve this character.
-\item But the topology generated by $D$ is finer than the topology generated by
-the Euclidian distance $\Delta$ -- which is the order topology.
-\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the
-chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order
-topology on $\mathds{R}$.
-\end{itemize}
-
-This result can be formulated as follows.
-
-\begin{theorem}
-\label{th:IC et topologie de l'ordre}
-The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the
-Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the
-order topology.
-\end{theorem}
-Indeed this result is weaker than the theorem establishing the chaos for the
-finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre}
-still remains important. Indeed, we have studied in our previous works a set
-different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$),
-in order to be as close as possible from the computer: the properties of
-disorder proved theoretically will then be preserved when computing. However, we
-could wonder whether this change does not lead to a disorder of a lower quality.
-In other words, have we replaced a situation of a good disorder lost when
-computing, to another situation of a disorder preserved but of bad quality.
-Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
-
+%\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}
+
+
+%\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.
+
+%\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.
+%
