X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/bce6fe373543bc9037b3de8504d9599882257bf5..02b8e8e685923bf2b12d529361c3435c0291337d:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 11dd246..150e434 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -34,7 +34,7 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
-\title{Efficient Generation of Pseudo-Random Bumbers based on Chaotic Iterations
+\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
 on GPU}
 \begin{document}
 
@@ -44,16 +44,99 @@ Guyeux\thanks{Authors in alphabetic order}}
 \maketitle
 
 \begin{abstract}
-This is the 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}
 
 \section{Introduction}
 
-Interet des itÃ©rations chaotiques pour gÃ©nÃ©rer des nombre alea\\
-Interet de gÃ©nÃ©rer des nombres alea sur GPU
-\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
-...
-
+Random  numbers are  used in  many scientific  applications and  simulations. On
+finite  state machines,  as computers,  it is  not possible  to  generate random
+numbers but only pseudo-random numbers. In practice, a good pseudo-random number
+generator (PRNG) needs  to verify some features to be used  by scientists. It is
+important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
+generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
+statistical properties. Finally, from our point a view, it is essential to prove
+that  a PRNG  is  chaotic.  Concerning  the  statistical tests,  TestU01 is  the
+best-known public-domain statistical testing package.   So we use it for all our
+PRNGs, especially the {\it BigCrush}  which provides the largest serie of tests.
+Concerning  the  chaotic properties,  Devaney~\cite{Devaney}  proposed a  common
+mathematical formulation of chaotic dynamical systems.
+
+In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic
+PRNG  based on  chaotic iterations. We  have proven  that these  PRNGs are
+chaotic in the Devaney's sense.  In this paper we propose a faster version which
+is also proven to be chaotic.
+
+Although graphics  processing units (GPU)  was initially designed  to accelerate
+the manipulation of  images, they are nowadays commonly  used in many scientific
+applications. Therefore,  it is important  to be able to  generate pseudo-random
+numbers inside a GPU when a scientific application runs in a GPU. That is why we
+also provide  an efficient  PRNG for  GPU respecting based  on IC.  Such devices
+allows us to generated almost 20 billions of random numbers per second.
+
+In order  to establish  that our  PRNGs are chaotic  according to  the Devaney's
+formulation, we  extend what we  have proposed in~\cite{guyeux10}.  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  relativitÃ© du  dÃ©sordre}  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.
+
+
+
+
+\section{Related works on GPU based PRNGs}
+\label{section:related works}
+In the litterature many authors have work on defining GPU based PRNGs. We do not
+want to be exhaustive and we just give the most significant works from our point
+of view. When authors mention the  number of random numbers generated per second
+we mention  it. We  consider that  a million numbers  per second  corresponds to
+1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
+
+In \cite{Pang:2008:cec},  the authors define  a PRNG based on  cellular automata
+which  does   not  require  high  precision  integer   arithmetics  nor  bitwise
+operations. There is no mention of statistical tests nor proof that this PRNG is
+chaotic.  Concerning   the  speed  of   generation,  they  can   generate  about
+3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
+
+In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
+based on  Lagged Fibonacci, Hybrid  Taus or Hybrid  Taus.  They have  used these
+PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
+GPU. Performance of  the GPU versions are far better than  those obtained with a
+CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
+no mention that their PRNGs have chaos mathematical properties.
+
+
+Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
+PRNGs on  diferrent computing architectures: CPU,  field-programmable gate array
+(FPGA), GPU and massively parallel  processor. This study is interesting because
+it  shows the  performance  of the  same  PRNGs on  different architeture.   For
+example,  the FPGA  is globally  the  fastest architecture  and it  is also  the
+efficient one because it provides the fastest number of generated random numbers
+per joule. Concerning the GPU,  authors can generate betweend 11 and 16GSample/s
+with a GTX 280  GPU. The drawback of this work is  that those PRNGs only succeed
+the {\it Crush} test which is easier than the {\it Big Crush} test.
+\newline
+\newline
+To the best of our knowledge no GPU implementation have been proven to have chaotic properties.
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
@@ -280,7 +363,7 @@ $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracke
 \rrbracket^{\mathds{N}}$ 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 
@@ -389,6 +472,7 @@ to the following discrete dynamical system in chaotic iterations:
   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
@@ -409,7 +493,7 @@ use of more general chaotic iterations to generate pseudo-random numbers
 faster, does not deflate their topological chaos properties.
 
 \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
-
+\label{deuxiÃ¨me def}
 Let us consider the discrete dynamical systems in chaotic iterations having 
 the general form:
 
@@ -623,15 +707,33 @@ claimed in the lemma.
 We can now prove the Theorem~\ref{t:chaos des general}...
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
- On the one hand, strong transitivity implies transitivity. On the other hand, 
-the regularity is exactly Lemma~\ref{strongTrans} with $Y=X$. As the sensitivity
-to the initial condition is implied by these two properties, we thus have
-the theorem.
+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
@@ -658,29 +760,7 @@ x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
  \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}
+
 
 
 
@@ -707,19 +787,19 @@ unsigned int CIprng() {
 
 
 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$>>$32)} selects the 32
-most  significants bits  of the  variable \texttt{t}.   So to  produce  a random
-number realizes  6 xor operations with  6 32-bits numbers produced  by 3 64-bits
-PRNG.  This version successes the  BigCrush of the TestU01 battery [P.  Lâecuyer
-  and R. Simard. Testu01].
-
-\section{Efficient prng based on chaotic iterations on GPU}
+based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
+This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
+\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
+xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
+each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$32)}  selects the 32 most significants  bits of the
+variable \texttt{t}.   So to produce a  random number realizes  6 xor operations
+with 6 32-bits  numbers produced by 3 64-bits PRNG.   This version successes the
+BigCrush of the TestU01 battery~\cite{LEcuyerS07}.
+
+\section{Efficient PRNGs based on chaotic iterations on GPU}
+\label{sec:efficient prng gpu}
 
 In  order to benefit  from computing  power of  GPU, a  program needs  to define
 independent blocks of threads which  can be computed simultaneously. In general,
@@ -727,8 +807,8 @@ 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  [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}
@@ -738,14 +818,14 @@ 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.
+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}
 
@@ -793,7 +873,7 @@ for all the differents nodes involves in the computation.
 
 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
+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
@@ -805,7 +885,7 @@ 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 succeed to the BigCrush batteries of tests.
+This version also succeeds to the {\it BigCrush} batteries of tests.
 
 \begin{algorithm}
 
@@ -816,17 +896,15 @@ tab1, tab2: Arrays containing permutations of size permutation\_size\;}
 
 \KwOut{NewNb: array containing random numbers in global memory}
 \If{threadId is concerned} {
-  retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
+  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]\;
   }
@@ -838,550 +916,597 @@ version}
 \label{algo:gpu_kernel2}
 \end{algorithm}
 
-
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in three operations having 
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, three iterations of the general chaotic
+iterations are realized between two stored values of the PRNG.
+To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+we must guarantee that this dynamical system iterates on the space 
+$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
+The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+To prevent from any flaws of chaotic properties, we must check that each right 
+term, corresponding to terms of the strategies,  can possibly be equal to any
+integer of $\llbracket 1, \mathsf{N} \rrbracket$. 
+
+Such a result is obvious for the two first lines, as for the xor-like(), all the
+integers belonging into its interval of definition can occur at each iteration.
+It can be easily stated for the two last lines by an immediate mathematical
+induction.
+
+Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
+chaotic iterations presented previously, and for this reason, it satisfies the 
+Devaney's formulation of a chaotic behavior.
 
 \section{Experiments}
-
-Differents 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}
+\label{fig:time_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,
-Ã©ventuellement en fonction du nombres de threads par bloc.
-
-
-
-\section{The relativity of disorder}
-\label{sec:de la relativitÃ© du dÃ©sordre}
-
-\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 $f$.
+In  comparison,   Listing~\ref{algo:seqCIprng}  allows  us   to  generate  about
+138MSample/s with only one core of the Xeon E5530.
 
-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{Cryptanalysis of the Proposed PRNG}
 
 
+Mettre ici la preuve de PCH
 
-\section{Chaos on the order topology}
+%\section{The relativity of disorder}
+%\label{sec:de la relativitÃ© du dÃ©sordre}
 
-\subsection{The phase space is an interval of the real line}
+%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{deuxiÃ¨me def}.
 
-\subsubsection{Toward a topological semiconjugacy}
+%\subsection{Impact of the topology's finenesse}
 
-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 us firstly introduce the following notations.
 
-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{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{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}
 
+%\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}
 
-$\varphi$ 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{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.
 
-\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}
+%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 $V$.
 
-We are now able to define the function $g$, whose goal is to translate the
-chaotic iterations $\Go$ on an interval of $\mathds{R}$.
+%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{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}
+%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}
 
-\bigskip
+%\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):
 
-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}
+%\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}
 
 
-\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}
+%\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.
 
-Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most
-usual one being the Euclidian distance recalled bellow:
+%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{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:
 
+%\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{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}
+%\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 $f$.
 
-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.
+%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}
 
 
-\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}
-
-
-\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 itÃ©rations 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.
- 
+%\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 itÃ©rations 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.
+% 
 
 
 
