X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/e4f33be63a2bad38b19204da8d8be5c7e2113e13..3abda4f59d76446238e6b891bc14e1ac7b44c34a:/prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 0ab28a1..f357476 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -14,9 +14,9 @@
 \usepackage{algorithmic}
 \usepackage{slashbox}
 \usepackage{ctable}
+\usepackage{cite}
 \usepackage{tabularx}
 \usepackage{multirow}
-
 % Pour mathds : les ensembles IR, IN, etc.
 \usepackage{dsfont}
 
@@ -26,6 +26,10 @@
 \usepackage{graphicx}
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
+\usepackage{xr-hyper}
+\usepackage{hyperref}
+\externaldocument[A-]{supplementary}
+
 
 
 \newtheorem{notation}{Notation}
@@ -51,8 +55,8 @@ Guyeux, and Pierre-Cyrille HÃ©am\thanks{Authors in alphabetic order}}
 \IEEEcompsoctitleabstractindextext{
 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
-graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations.  It
-is firstly proven  to be chaotic according to the Devaney's  formulation. We thus propose  an efficient
+graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations and
+it is thus chaotic according to the Devaney's  formulation. We propose  an efficient
 implementation  for  GPU that successfully passes the   {\it BigCrush} tests, deemed to be the  hardest
 battery of tests in TestU01.  Experiments show that this PRNG can generate
 about 20 billion of random numbers  per second on Tesla C1060 and NVidia GTX280
@@ -188,12 +192,11 @@ The remainder of this paper  is organized as follows. In Section~\ref{section:re
   and on an iteration process called ``chaotic
 iterations'' on which the post-treatment is based. 
 The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.
-
-Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
-improvement related to the chaotic iteration based post-treatment, for
-our previously released PRNGs and a new efficient 
+Section~\ref{sec:efficient PRNG} %{The generation of pseudorandom sequence} %illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and
+ contains a new efficient 
 implementation on CPU.
-
  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
@@ -201,8 +204,8 @@ Section~\ref{sec:experiments}.
 We show in Section~\ref{sec:security analysis} that, if the inputted
 generator is cryptographically secure, then it is the case too for the
 generator provided by the post-treatment.
-A practical
-security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
+%A practical
+%security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
 Such a proof leads to the proposition of a cryptographically secure and
 chaotic generator on GPU based on the famous Blum Blum Shub
 in Section~\ref{sec:CSGPU} and to an improvement of the
@@ -715,583 +718,605 @@ in what follows).
 However, proofs
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
-\ref{Def:chaotic iterations}. The question is now to determine whether the
+\ref{Def:chaotic iterations}. The question to determine whether the
 use of more general chaotic iterations to generate pseudorandom numbers 
-faster, does not deflate their topological chaos properties.
+faster, does not deflate their topological chaos properties, has been
+investigated in Annex~\ref{A-deuxiÃ¨me def}, leading to the following result.
 
-\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: $\forall    n\in     \mathds{N}^{\ast     }$, $  \forall     i\in
-\llbracket1;\mathsf{N}\rrbracket $,
+ \begin{theorem}
+ \label{t:chaos des general}
+  The general chaotic iterations defined in Equation~\ref{eq:generalIC}
+satisfy
+ the Devaney's property of chaos.
+ \end{theorem}
 
-\begin{equation}
-  x_i^n=\left\{
-\begin{array}{ll}
-  x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
-  \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
-\end{array}\right.
-\label{general CIs}
-\end{equation}
 
-In other words, at the $n^{th}$ iteration, only the cells whose id is
-contained into the set $S^{n}$ are iterated.
+%%RAF proof en supplementary, j'ai mis le theorem.
+% A vÃ©rifier
 
-Let us now rewrite these general chaotic iterations as usual discrete dynamical
-system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
-is required in order to study the topological behavior of the system.
+% \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+%\label{deuxiÃ¨me def}
+%The proof is given in Section~\ref{A-deuxiÃ¨me def} of the annex document.
+%% \label{deuxiÃ¨me def}
+%% Let us consider the discrete dynamical systems in chaotic iterations having 
+%% the general form: $\forall    n\in     \mathds{N}^{\ast     }$, $  \forall     i\in
+%% \llbracket1;\mathsf{N}\rrbracket $,
 
-Let us introduce the following function:
-\begin{equation}
-\begin{array}{cccc}
- \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
-         & (i,X) & \longmapsto  & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X,  \end{array}\right.
-\end{array} 
-\end{equation}
-where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
+%% \begin{equation}
+%%   x_i^n=\left\{
+%% \begin{array}{ll}
+%%   x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
+%%   \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+%% \end{array}\right.
+%% \label{general CIs}
+%% \end{equation}
 
-Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
-$F_{f}:  \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} 
-\longrightarrow \mathds{B}^{\mathsf{N}}$
-\begin{equation*}
-\begin{array}{rll}
- (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
-\end{array}%
-\end{equation*}%
-where + and . are the Boolean addition and product operations, and $\overline{x}$ 
-is the negation of the Boolean $x$.
-Consider the phase space:
-\begin{equation}
-\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
-\mathds{B}^\mathsf{N},
-\end{equation}
-\noindent and the map defined on $\mathcal{X}$:
-\begin{equation}
-G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai virÃ© ce label qui existe dÃ©jÃ  avant...
-\end{equation}
-\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
-(S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
-\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function} 
-$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$. 
-Then the general chaotic iterations defined in Equation \ref{general CIs} can 
-be described by the following discrete dynamical system:
-\begin{equation}
-\left\{
-\begin{array}{l}
-X^0 \in \mathcal{X} \\
-X^{k+1}=G_{f}(X^k).%
-\end{array}%
-\right.
-\end{equation}%
+%% In other words, at the $n^{th}$ iteration, only the cells whose id is
+%% contained into the set $S^{n}$ are iterated.
 
-Once more, a shift function appears as a component of these general chaotic 
-iterations. 
+%% Let us now rewrite these general chaotic iterations as usual discrete dynamical
+%% system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
+%% is required in order to study the topological behavior of the system.
 
-To study the Devaney's chaos property, a distance between two points 
-$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
-Let us introduce:
-\begin{equation}
-d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
-\label{nouveau d}
-\end{equation}
-\noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}%
- }\delta (E_{k},\check{E}_{k})}$  is once more the Hamming distance, and
-$  \displaystyle{d_{s}(S,\check{S})}  =  \displaystyle{\dfrac{9}{\mathsf{N}}%
- \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$,
-%%RAPH : ici, j'ai supprimÃ© tous les sauts Ã  la ligne
+%% Let us introduce the following function:
+%% \begin{equation}
+%% \begin{array}{cccc}
+%%  \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
+%%          & (i,X) & \longmapsto  & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X,  \end{array}\right.
+%% \end{array} 
+%% \end{equation}
+%% where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
+
+%% Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
+%% $F_{f}:  \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} 
+%% \longrightarrow \mathds{B}^{\mathsf{N}}$
+%% \begin{equation*}
+%% \begin{array}{rll}
+%%  (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
+%% \end{array}%
+%% \end{equation*}%
+%% where + and . are the Boolean addition and product operations, and $\overline{x}$ 
+%% is the negation of the Boolean $x$.
+%% Consider the phase space:
+%% \begin{equation}
+%% \mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
+%% \mathds{B}^\mathsf{N},
+%% \end{equation}
+%% \noindent and the map defined on $\mathcal{X}$:
+%% \begin{equation}
+%% G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai virÃ© ce label qui existe dÃ©jÃ  avant...
+%% \end{equation}
+%% \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
+%% (S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
+%% \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function} 
+%% $i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$. 
+%% Then the general chaotic iterations defined in Equation \ref{general CIs} can 
+%% be described by the following discrete dynamical system:
 %% \begin{equation}
 %% \left\{
-%% \begin{array}{lll}
-%% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
-%% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\
-%% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
-%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
+%% \begin{array}{l}
+%% X^0 \in \mathcal{X} \\
+%% X^{k+1}=G_{f}(X^k).%
 %% \end{array}%
 %% \right.
-%% \end{equation}
-where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
-$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
-
-
-\begin{proposition}
-The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
-\end{proposition}
-
-\begin{proof}
- $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
-too, thus $d$, as being the sum of two distances, will also be a distance.
- \begin{itemize}
-\item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
-$d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
-$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
- \item $d_s$ is symmetric 
-($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
-of the symmetric difference. 
-\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, 
-and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, 
-we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
-inequality is obtained.
- \end{itemize}
-\end{proof}
-
-
-Before being able to study the topological behavior of the general 
-chaotic iterations, we must first establish that:
-
-\begin{proposition}
- For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 
-$\left( \mathcal{X},d\right)$.
-\end{proposition}
-
-
-\begin{proof}
-We use the sequential continuity.
-Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
-\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
-G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
-G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
-thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
-sequences).\newline
-As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
-to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
-d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
-In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
-cell will change its state:
-$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
-
-In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
-\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
-n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
-first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
-
-Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
-identical and strategies $S^n$ and $S$ start with the same first term.\newline
-Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
-so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
-\noindent We now prove that the distance between $\left(
-G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
-0. Let $\varepsilon >0$. \medskip
-\begin{itemize}
-\item If $\varepsilon \geqslant 1$, we see that the distance
-between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
-strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
-\medskip
-\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
-\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
-\begin{equation*}
-\exists n_{2}\in \mathds{N},\forall n\geqslant
-n_{2},d_{s}(S^n,S)<10^{-(k+2)},
-\end{equation*}%
-thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
-\end{itemize}
-\noindent As a consequence, the $k+1$ first entries of the strategies of $%
-G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
-the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
-10^{-(k+1)}\leqslant \varepsilon $.
-
-In conclusion,
-%%RAPH : ici j'ai rajoutÃ© une ligne
-%%TOF : ici j'ai rajoutÃ© un commentaire
-%%TOF : ici aussi
-$
-\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
-,$ $\forall n\geqslant N_{0},$
-$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
-\leqslant \varepsilon .
-$
-$G_{f}$ is consequently continuous.
-\end{proof}
-
-
-It is now possible to study the topological behavior of the general chaotic
-iterations. We will prove that,
+%% \end{equation}%
 
-\begin{theorem}
-\label{t:chaos des general}
- The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
-the Devaney's property of chaos.
-\end{theorem}
-
-Let us firstly prove the following lemma.
-
-\begin{lemma}[Strong transitivity]
-\label{strongTrans}
- For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
-find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
-\end{lemma}
-
-\begin{proof}
- Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
-Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
-are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
-$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
-We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
-that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
-the form $(S',E')$ where $E'=E$ and $S'$ starts with 
-$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
-\begin{itemize}
- \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
- \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
-\end{itemize}
-Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
-where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
-claimed in the lemma.
-\end{proof}
+%% Once more, a shift function appears as a component of these general chaotic 
+%% iterations. 
 
-We can now prove the Theorem~\ref{t:chaos des general}.
-
-\begin{proof}[Theorem~\ref{t:chaos des general}]
-Firstly, strong transitivity implies transitivity.
-
-Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
-prove that $G_f$ is regular, it is sufficient to prove that
-there exists a strategy $\tilde S$ such that the distance between
-$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
-$(\tilde S,E)$ is a periodic point.
-
-Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
-configuration that we obtain from $(S,E)$ after $t_1$ iterations of
-$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
-and $t_2\in\mathds{N}$ such
-that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
-
-Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
-of $S$ and the first $t_2$ terms of $S'$: 
-%%RAPH : j'ai coupÃ© la ligne en 2
-$$\tilde
-S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ 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{Statistical Improvements Using Chaotic Iterations}
-
-\label{The generation of pseudorandom sequence}
-
-
-Let us now explain why we have reasonable ground to believe that chaos 
-can improve statistical properties.
-We will show in this section that chaotic properties as defined in the
-mathematical theory of chaos are related to some statistical tests that can be found
-in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
-chaotic iterations, the new generator presents better statistical properties
-(this section summarizes and extends the work of~\cite{bfg12a:ip}).
+%% To study the Devaney's chaos property, a distance between two points 
+%% $X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
+%% Let us introduce:
+%% \begin{equation}
+%% d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
+%% \label{nouveau d}
+%% \end{equation}
+%% \noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}%
+%%  }\delta (E_{k},\check{E}_{k})}$  is once more the Hamming distance, and
+%% $  \displaystyle{d_{s}(S,\check{S})}  =  \displaystyle{\dfrac{9}{\mathsf{N}}%
+%%  \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$,
+%% %%RAPH : ici, j'ai supprimÃ© tous les sauts Ã  la ligne
+%% %% \begin{equation}
+%% %% \left\{
+%% %% \begin{array}{lll}
+%% %% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
+%% %% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\
+%% %% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
+%% %% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
+%% %% \end{array}%
+%% %% \right.
+%% %% \end{equation}
+%% where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
+%% $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
+
+
+%% \begin{proposition}
+%% The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
+%% \end{proposition}
+
+%% \begin{proof}
+%%  $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
+%% too, thus $d$, as being the sum of two distances, will also be a distance.
+%%  \begin{itemize}
+%% \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
+%% $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
+%% $\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
+%%  \item $d_s$ is symmetric 
+%% ($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
+%% of the symmetric difference. 
+%% \item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, 
+%% and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, 
+%% we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
+%% inequality is obtained.
+%%  \end{itemize}
+%% \end{proof}
+
+
+%% Before being able to study the topological behavior of the general 
+%% chaotic iterations, we must first establish that:
+
+%% \begin{proposition}
+%%  For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 
+%% $\left( \mathcal{X},d\right)$.
+%% \end{proposition}
+
+
+%% \begin{proof}
+%% We use the sequential continuity.
+%% Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
+%% \mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
+%% G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
+%% G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
+%% thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
+%% sequences).\newline
+%% As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
+%% to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
+%% d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
+%% In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
+%% cell will change its state:
+%% $\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
+
+%% In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
+%% \mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
+%% n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
+%% first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
+
+%% Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
+%% identical and strategies $S^n$ and $S$ start with the same first term.\newline
+%% Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
+%% so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
+%% \noindent We now prove that the distance between $\left(
+%% G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
+%% 0. Let $\varepsilon >0$. \medskip
+%% \begin{itemize}
+%% \item If $\varepsilon \geqslant 1$, we see that the distance
+%% between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
+%% strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
+%% \medskip
+%% \item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
+%% \varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
+%% \begin{equation*}
+%% \exists n_{2}\in \mathds{N},\forall n\geqslant
+%% n_{2},d_{s}(S^n,S)<10^{-(k+2)},
+%% \end{equation*}%
+%% thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
+%% \end{itemize}
+%% \noindent As a consequence, the $k+1$ first entries of the strategies of $%
+%% G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
+%% the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
+%% 10^{-(k+1)}\leqslant \varepsilon $.
+
+%% In conclusion,
+%% %%RAPH : ici j'ai rajoutÃ© une ligne
+%% %%TOF : ici j'ai rajoutÃ© un commentaire
+%% %%TOF : ici aussi
+%% $
+%% \forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+%% ,$ $\forall n\geqslant N_{0},$
+%% $ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+%% \leqslant \varepsilon .
+%% $
+%% $G_{f}$ is consequently continuous.
+%% \end{proof}
+
+
+%% It is now possible to study the topological behavior of the general chaotic
+%% iterations. We will prove that,
+
+%% \begin{theorem}
+%% \label{t:chaos des general}
+%%  The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+%% the Devaney's property of chaos.
+%% \end{theorem}
+
+%% Let us firstly prove the following lemma.
+
+%% \begin{lemma}[Strong transitivity]
+%% \label{strongTrans}
+%%  For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
+%% find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+%% \end{lemma}
+
+%% \begin{proof}
+%%  Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
+%% Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
+%% are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
+%% $\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+%% We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+%% that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+%% the form $(S',E')$ where $E'=E$ and $S'$ starts with 
+%% $(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+%% \begin{itemize}
+%%  \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+%%  \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+%% \end{itemize}
+%% Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
+%% where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+%% claimed in the lemma.
+%% \end{proof}
+
+%% We can now prove the Theorem~\ref{t:chaos des general}.
+
+%% \begin{proof}[Theorem~\ref{t:chaos des general}]
+%% Firstly, strong transitivity implies transitivity.
+
+%% Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+%% prove that $G_f$ is regular, it is sufficient to prove that
+%% there exists a strategy $\tilde S$ such that the distance between
+%% $(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+%% $(\tilde S,E)$ is a periodic point.
+
+%% Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+%% configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+%% $G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
+%% and $t_2\in\mathds{N}$ such
+%% that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+%% Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+%% of $S$ and the first $t_2$ terms of $S'$: 
+%% %%RAPH : j'ai coupÃ© la ligne en 2
+%% $$\tilde
+%% S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ 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}
+
+
+
+
+%%RAF : mis en supplementary
+
+
+%\section{Statistical Improvements Using Chaotic Iterations}
+%\label{The generation of pseudorandom sequence}
+%The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
+The reasons to desire chaos to achieve randomness are given in Annex~\ref{A-The generation of pseudorandom sequence}.
+
+%% \label{The generation of pseudorandom sequence}
+
+
+%% Let us now explain why we have reasonable ground to believe that chaos 
+%% can improve statistical properties.
+%% We will show in this section that chaotic properties as defined in the
+%% mathematical theory of chaos are related to some statistical tests that can be found
+%% in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
+%% chaotic iterations, the new generator presents better statistical properties
+%% (this section summarizes and extends the work of~\cite{bfg12a:ip}).
+
+
+
+%% \subsection{Qualitative relations between topological properties and statistical tests}
+
+
+%% There are various relations between topological properties that describe an unpredictable behavior for a discrete 
+%% dynamical system on the one
+%% hand, and statistical tests to check the randomness of a numerical sequence
+%% on the other hand. These two mathematical disciplines follow a similar 
+%% objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
+%% recurrent sequence), with two different but complementary approaches.
+%% It is true that the following illustrative links give only qualitative arguments, 
+%% and proofs should be provided later to make such arguments irrefutable. However 
+%% they give a first understanding of the reason why we think that chaotic properties should tend
+%% to improve the statistical quality of PRNGs.
+%% %
+%% Let us now list some of these relations between topological properties defined in the mathematical
+%% theory of chaos and tests embedded into the NIST battery. %Such relations need to be further 
+%% %investigated, but they presently give a first illustration of a trend to search similar properties in the 
+%% %two following fields: mathematical chaos and statistics.
+
+
+%% \begin{itemize}
+%%     \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must 
+%% have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
+%% a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
+%% is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
+%% knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
+%% the two following NIST tests~\cite{Nist10}:
+%%     \begin{itemize}
+%%         \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
+%%         \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
+%%     \end{itemize}
+
+%% \item \textbf{Transitivity}. This topological property previously introduced  states that the dynamical system is intrinsically complicated: it cannot be simplified into 
+%% two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space. 
+%% This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
+%% of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention 
+%% is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
+%%     \begin{itemize}
+%%         \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
+%%         \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
+%%     \end{itemize}
+
+%% \item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
+%% to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
+%%     \begin{itemize}
+%%         \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
+%%     \end{itemize}
+%%     \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
+%% has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
+%% rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach, 
+%% whereas topological entropy is defined as follows:
+%% $x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
+%% leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations, 
+%% the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f)  = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
+%% This value measures the average exponential growth of the number of distinguishable orbit segments. 
+%% In this sense, it measures the complexity of the topological dynamical system, whereas 
+%% the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
+%%     \begin{itemize}
+%% \item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
+%%     \end{itemize}
+
+%%     \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are 
+%% not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
+%%     \begin{itemize}
+%% \item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
+%% \item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
+%%       \end{itemize}
+%% \end{itemize}
+
+
+%% We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:CaractÃ©risation   des   IC   chaotiques} are, among other
+%% things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
+%% and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
+%% where $\mathsf{N}$ is the size of the iterated vector.
+%% These topological properties make that we are ground to believe that a generator based on chaotic
+%% iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
+%% the NIST one. The following subsections, in which we prove that defective generators have their
+%% statistical properties improved by chaotic iterations, show that such an assumption is true.
+
+%% \subsection{Details of some Existing Generators}
+
+%% The list of defective PRNGs we will use 
+%% as inputs for the statistical tests to come is introduced here.
+
+%% Firstly, the simple linear congruency generators (LCGs) will be used. 
+%% They are defined by the following recurrence:
+%% \begin{equation}
+%% x^n = (ax^{n-1} + c)~mod~m,
+%% \label{LCG}
+%% \end{equation}
+%% where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to 
+%% $m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three) 
+%% combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
 
+%% Secondly, the multiple recursive generators (MRGs) which will be used,
+%% are based on a linear recurrence of order 
+%% $k$, modulo $m$~\cite{LEcuyerS07}:
+%% \begin{equation}
+%% x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .
+%% \label{MRG}
+%% \end{equation}
+%% The combination of two MRGs (referred as 2MRGs) is also used in these experiments.
 
+%% Generators based on linear recurrences with carry will be regarded too.
+%% This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
+%% \begin{equation}
+%% \label{AWC}
+%% \begin{array}{l}
+%% x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
+%% c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
+%% the SWB generator, having the recurrence:
+%% \begin{equation}
+%% \label{SWB}
+%% \begin{array}{l}
+%% x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
+%% c^n=\left\{
+%% \begin{array}{l}
+%% 1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
+%% 0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
+%% and the SWC generator, which is based on the following recurrence:
+%% \begin{equation}
+%% \label{SWC}
+%% \begin{array}{l}
+%% x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
+%% c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
 
-\subsection{Qualitative relations between topological properties and statistical tests}
+%% Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
+%% \begin{equation}
+%% x^n = x^{n-r} \oplus x^{n-k} .
+%% \label{GFSR}
+%% \end{equation}
 
 
-There are various relations between topological properties that describe an unpredictable behavior for a discrete 
-dynamical system on the one
-hand, and statistical tests to check the randomness of a numerical sequence
-on the other hand. These two mathematical disciplines follow a similar 
-objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
-recurrent sequence), with two different but complementary approaches.
-It is true that the following illustrative links give only qualitative arguments, 
-and proofs should be provided later to make such arguments irrefutable. However 
-they give a first understanding of the reason why we think that chaotic properties should tend
-to improve the statistical quality of PRNGs.
-%
-Let us now list some of these relations between topological properties defined in the mathematical
-theory of chaos and tests embedded into the NIST battery. %Such relations need to be further 
-%investigated, but they presently give a first illustration of a trend to search similar properties in the 
-%two following fields: mathematical chaos and statistics.
+%% Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
 
+%% \begin{equation}
+%% \label{INV}
+%% \begin{array}{l}
+%% x^n=\left\{
+%% \begin{array}{ll}
+%% (a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
+%% a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
 
-\begin{itemize}
-    \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must 
-have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
-a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
-is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
-knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
-the two following NIST tests~\cite{Nist10}:
-    \begin{itemize}
-        \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
-        \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
-    \end{itemize}
-
-\item \textbf{Transitivity}. This topological property previously introduced  states that the dynamical system is intrinsically complicated: it cannot be simplified into 
-two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space. 
-This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
-of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention 
-is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
-    \begin{itemize}
-        \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
-        \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
-    \end{itemize}
-
-\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
-to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
-    \begin{itemize}
-        \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
-    \end{itemize}
-    \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
-has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
-rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach, 
-whereas topological entropy is defined as follows:
-$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
-leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations, 
-the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f)  = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
-This value measures the average exponential growth of the number of distinguishable orbit segments. 
-In this sense, it measures the complexity of the topological dynamical system, whereas 
-the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
-    \begin{itemize}
-\item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
-    \end{itemize}
-
-    \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are 
-not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
-    \begin{itemize}
-\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
-\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
-      \end{itemize}
-\end{itemize}
 
 
-We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:CaractÃ©risation   des   IC   chaotiques} are, among other
-things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
-and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
-where $\mathsf{N}$ is the size of the iterated vector.
-These topological properties make that we are ground to believe that a generator based on chaotic
-iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
-the NIST one. The following subsections, in which we prove that defective generators have their
-statistical properties improved by chaotic iterations, show that such an assumption is true.
+%% \begin{table}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{TestU01 Statistical Test Failures}
+%% \label{TestU011}
+%% \centering
+%%   \begin{tabular}{lccccc}
+%%     \toprule
+%% Test name &Tests& Logistic 		& XORshift	& ISAAC\\
+%% Rabbit 				& 	38	&21  	 	&14	&0	 \\
+%% Alphabit 			&	17	&16 		&9	&0	 \\
+%% Pseudo DieHARD 			&126	&0 	 	&2	&0	\\
+%% FIPS\_140\_2 			&16	&0 		&0	&0	\\
+%% SmallCrush 			&15	&4 		&5	&0	 \\
+%% Crush 				&144	&95 		&57	&0	 \\
+%% Big Crush 			&160	&125 	 	&55	&0	 \\ \hline
+%% Failures 		&	&261 	 	&146	&0	 \\
+%% \bottomrule
+%%   \end{tabular}
+%% \end{table}
 
-\subsection{Details of some Existing Generators}
 
-The list of defective PRNGs we will use 
-as inputs for the statistical tests to come is introduced here.
 
-Firstly, the simple linear congruency generators (LCGs) will be used. 
-They are defined by the following recurrence:
-\begin{equation}
-x^n = (ax^{n-1} + c)~mod~m,
-\label{LCG}
-\end{equation}
-where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to 
-$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three) 
-combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
+%% \begin{table}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
+%% \label{TestU01 for Old CI}
+%% \centering
+%%   \begin{tabular}{lcccc}
+%%     \toprule
+%% \multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
+%% &Logistic& XORshift& ISAAC&ISAAC  \\ 
+%% &+& +& + & + \\ 
+%% &Logistic& XORshift& XORshift&ISAAC  \\ \cmidrule(r){2-5}
+%% Rabbit 				  	&7 	&2 	&0 	&0	 \\
+%% Alphabit 			 	& 3	&0 	&0 	&0	 \\
+%% DieHARD 		 	&0 	&0 	&0 	&0	\\
+%% FIPS\_140\_2 		 	&0 	&0	&0 	&0	\\
+%% SmallCrush 			 	&2 	&0	&0 	&0	 \\
+%% Crush 				 	&47 	&4 	&0 	&0	 \\
+%% Big Crush 			 	&79 	&3 	&0	&0	 \\ \hline
+%% Failures 	 		 	&138 	&9 	&0 	&0	 \\
+%% \bottomrule
+%%   \end{tabular}
+%% \end{table}
 
-Secondly, the multiple recursive generators (MRGs) which will be used,
-are based on a linear recurrence of order 
-$k$, modulo $m$~\cite{LEcuyerS07}:
-\begin{equation}
-x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .
-\label{MRG}
-\end{equation}
-The combination of two MRGs (referred as 2MRGs) is also used in these experiments.
 
-Generators based on linear recurrences with carry will be regarded too.
-This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
-\begin{equation}
-\label{AWC}
-\begin{array}{l}
-x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
-c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
-the SWB generator, having the recurrence:
-\begin{equation}
-\label{SWB}
-\begin{array}{l}
-x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
-c^n=\left\{
-\begin{array}{l}
-1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
-0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
-and the SWC generator, which is based on the following recurrence:
-\begin{equation}
-\label{SWC}
-\begin{array}{l}
-x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
-c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
-
-Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
-\begin{equation}
-x^n = x^{n-r} \oplus x^{n-k} .
-\label{GFSR}
-\end{equation}
 
 
-Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
 
-\begin{equation}
-\label{INV}
-\begin{array}{l}
-x^n=\left\{
-\begin{array}{ll}
-(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
-a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
-
-
-
-\begin{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test Failures}
-\label{TestU011}
-\centering
-  \begin{tabular}{lccccc}
-    \toprule
-Test name &Tests& Logistic 		& XORshift	& ISAAC\\
-Rabbit 				& 	38	&21  	 	&14	&0	 \\
-Alphabit 			&	17	&16 		&9	&0	 \\
-Pseudo DieHARD 			&126	&0 	 	&2	&0	\\
-FIPS\_140\_2 			&16	&0 		&0	&0	\\
-SmallCrush 			&15	&4 		&5	&0	 \\
-Crush 				&144	&95 		&57	&0	 \\
-Big Crush 			&160	&125 	 	&55	&0	 \\ \hline
-Failures 		&	&261 	 	&146	&0	 \\
-\bottomrule
-  \end{tabular}
-\end{table}
-
-
-
-\begin{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
-\label{TestU01 for Old CI}
-\centering
-  \begin{tabular}{lcccc}
-    \toprule
-\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
-&Logistic& XORshift& ISAAC&ISAAC  \\ 
-&+& +& + & + \\ 
-&Logistic& XORshift& XORshift&ISAAC  \\ \cmidrule(r){2-5}
-Rabbit 				  	&7 	&2 	&0 	&0	 \\
-Alphabit 			 	& 3	&0 	&0 	&0	 \\
-DieHARD 		 	&0 	&0 	&0 	&0	\\
-FIPS\_140\_2 		 	&0 	&0	&0 	&0	\\
-SmallCrush 			 	&2 	&0	&0 	&0	 \\
-Crush 				 	&47 	&4 	&0 	&0	 \\
-Big Crush 			 	&79 	&3 	&0	&0	 \\ \hline
-Failures 	 		 	&138 	&9 	&0 	&0	 \\
-\bottomrule
-  \end{tabular}
-\end{table}
-
-
-
-
-
-\subsection{Statistical tests}
-\label{Security analysis}
-
-Three batteries of tests are reputed and regularly used
-to evaluate the statistical properties of newly designed pseudorandom
-number generators. These batteries are named DieHard~\cite{Marsaglia1996},
-the NIST suite~\cite{ANDREW2008}, and the most stringent one called
-TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
-
-
-
-\label{Results and discussion}
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
-\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
-\centering
-  \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
-    \hline\hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB  & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
-NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15}   & 14/15 & 14/15  & 14/15 & 14/15& 14/15& 14/15 \\ \hline
-DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
-
-Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
-results on the two first batteries recalled above, indicating that all the PRNGs presented
-in the previous section
-cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot 
-fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
-iterations can solve this issue.
-%More precisely, to
-%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
-%\begin{enumerate}
-%  \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
-%  \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
-%  \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
-%\begin{equation}
-%\begin{array}{l}
-%\left\{
-%\begin{array}{l}
-%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
-%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
-%\end{equation}
-%$m$ is called the \emph{functional power}.
-%\end{enumerate}
-%
-The obtained results are reproduced in Table
-\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
-The scores written in boldface indicate that all the tests have been passed successfully, whereas an 
-asterisk ``*'' means that the considered passing rate has been improved.
-The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
-Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
- are not as good as for the two other versions of these CIPRNGs.
-However 8 tests have been improved (with no deflation for the other results).
-
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
-\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
-\centering
-  \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
-    \hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG  & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
-Old CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}   & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} *  & \textbf{18/18} *  & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
-New CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}  & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
-Xor CIPRNG\\ \hline\hline
-NIST & 14/15*& \textbf{15/15} *   & \textbf{15/15}   & \textbf{15/15}   & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15}  \\ \hline
-DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
-
-
-We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
-the statistical behavior of the Xor CI version by combining more than one 
-$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
-the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
-Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained 
-using chaotic iterations on defective generators.
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
-\label{threshold}
-\centering
-  \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
-    \hline
-Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3  & MRG2 \\ \hline\hline
-Threshold  value $m$& 19 & 7  & 2& 1 & 11& 9& 3& 4\\ \hline\hline
-\end{tabular}
-\end{table*}
-
-Finally, the TestU01 battery has been launched on three well-known generators 
-(a logistic map, a simple XORshift, and the cryptographically secure ISAAC, 
-see Table~\ref{TestU011}). These results can be compared with 
-Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
-Old CI PRNG that has received these generators.
-The obvious improvement speaks for itself, and together with the other
-results recalled in this section, it reinforces the opinion that a strong
-correlation between topological properties and statistical behavior exists.
-
-
-The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
-fastest generator in the chaotic iteration based family. In the remainder,
-this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
-raise ambiguity.
-
-
-\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
+%% \subsection{Statistical tests}
+%% \label{Security analysis}
+
+%% Three batteries of tests are reputed and regularly used
+%% to evaluate the statistical properties of newly designed pseudorandom
+%% number generators. These batteries are named DieHard~\cite{Marsaglia1996},
+%% the NIST suite~\cite{ANDREW2008}, and the most stringent one called
+%% TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
+
+
+
+%% \label{Results and discussion}
+%% \begin{table*}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
+%% \label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
+%% \centering
+%%   \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
+%%     \hline\hline
+%% Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%% \backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB  & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
+%% NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15}   & 14/15 & 14/15  & 14/15 & 14/15& 14/15& 14/15 \\ \hline
+%% DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
+%% \end{tabular}
+%% \end{table*}
+
+%% Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
+%% results on the two first batteries recalled above, indicating that all the PRNGs presented
+%% in the previous section
+%% cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot 
+%% fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
+%% iterations can solve this issue.
+%% %More precisely, to
+%% %illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
+%% %\begin{enumerate}
+%% %  \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
+%% %  \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
+%% %  \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%% %\begin{equation}
+%% %\begin{array}{l}
+%% %\left\{
+%% %\begin{array}{l}
+%% %x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
+%% %\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
+%% %\end{equation}
+%% %$m$ is called the \emph{functional power}.
+%% %\end{enumerate}
+%% %
+%% The obtained results are reproduced in Table
+%% \ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
+%% The scores written in boldface indicate that all the tests have been passed successfully, whereas an 
+%% asterisk ``*'' means that the considered passing rate has been improved.
+%% The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
+%% Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
+%%  are not as good as for the two other versions of these CIPRNGs.
+%% However 8 tests have been improved (with no deflation for the other results).
+
+
+%% \begin{table*}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
+%% \label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
+%% \centering
+%%   \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
+%%     \hline
+%% Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%% \backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG  & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
+%% Old CIPRNG\\ \hline \hline
+%% NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}   & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%% DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} *  & \textbf{18/18} *  & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
+%% New CIPRNG\\ \hline \hline
+%% NIST & \textbf{15/15} *  & \textbf{15/15} * & \textbf{15/15}   & \textbf{15/15}  & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%% DieHARD & \textbf{18/18} *  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
+%% Xor CIPRNG\\ \hline\hline
+%% NIST & 14/15*& \textbf{15/15} *   & \textbf{15/15}   & \textbf{15/15}   & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15}  \\ \hline
+%% DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18}  & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
+%% \end{tabular}
+%% \end{table*}
+
+
+%% We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
+%% the statistical behavior of the Xor CI version by combining more than one 
+%% $\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
+%% the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
+%% Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained 
+%% using chaotic iterations on defective generators.
+
+%% \begin{table*}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
+%% \label{threshold}
+%% \centering
+%%   \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
+%%     \hline
+%% Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3  & MRG2 \\ \hline\hline
+%% Threshold  value $m$& 19 & 7  & 2& 1 & 11& 9& 3& 4\\ \hline\hline
+%% \end{tabular}
+%% \end{table*}
+
+%% Finally, the TestU01 battery has been launched on three well-known generators 
+%% (a logistic map, a simple XORshift, and the cryptographically secure ISAAC, 
+%% see Table~\ref{TestU011}). These results can be compared with 
+%% Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
+%% Old CI PRNG that has received these generators.
+%% The obvious improvement speaks for itself, and together with the other
+%% results recalled in this section, it reinforces the opinion that a strong
+%% correlation between topological properties and statistical behavior exists.
+
+
+%% The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
+%% fastest generator in the chaotic iteration based family. In the remainder,
+%% this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
+%% raise ambiguity.
+
+
+\section{First Efficient Implementation of a PRNG based on Chaotic Iterations}
 \label{sec:efficient PRNG}
 %
 %Based on the proof presented in the previous section, it is now possible to 
@@ -1613,13 +1638,13 @@ as it is shown in the next sections.
 
 
 This section is dedicated to the security analysis of the
-  proposed PRNGs, both from a theoretical and from a practical point of view.
+  proposed PRNGs.%, both from a theoretical and from a practical point of view.
 
-\subsection{Theoretical Proof of Security}
+%\subsection{Theoretical Proof of Security}
 \label{sec:security analysis}
 
 The standard definition
-  of {\it indistinguishability} used is the classical one as defined for
+  of {\it indistinguishability} used here is the classical one as defined for
   instance in~\cite[chapter~3]{Goldreich}. 
   This property shows that predicting the future results of the PRNG
   cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this
@@ -1628,7 +1653,7 @@ The standard definition
   be broken in practice. But it also means that if the keys/seeds are large
   enough, the system is secured.
 As a complement, an example of a concrete practical evaluation of security
-is outlined in the next subsection.
+is outlined in Annex~\ref{A-sec:Practicak evaluation}.
 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
@@ -1741,94 +1766,99 @@ proving that $H$ is not secure, which is a contradiction.
 
 
 
-\subsection{Practical Security Evaluation}
-\label{sec:Practicak evaluation}
+%\subsection{Practical Security Evaluation}
+%\label{sec:Practicak evaluation}
+%This subsection is given in Section
+A example of a practical security evaluation is outlined in
+Annex~\ref{A-sec:Practicak evaluation}.
+%%RAF mis en annexe
 
-Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when
-they are XORed with an already cryptographically
-secure PRNG. But, as stated previously,
-such a property does not mean that, whatever the
-key size, no attacker can predict the next bit
-knowing all the previously released ones.
-However, given a key size, it is possible to 
-measure in practice the minimum duration needed
-for an attacker to break a cryptographically
-secure PRNG, if we know the power of his/her
-machines. Such a concrete security evaluation 
-is related to the $(T,\varepsilon)-$security
-notion, which is recalled and evaluated in what 
-follows, for the sake of completeness.
 
-Let us firstly recall that,
-\begin{definition}
-Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
-in time $T$. 
-Let $\varepsilon > 0$. 
-$\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
-generator $G$ if
-
-\begin{flushleft}
-$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
-\end{flushleft}
-
-\begin{flushright}
-$ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
-\end{flushright}
-
-\noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
-``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
-corresponding set.
-\end{definition}
+%% Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when
+%% they are XORed with an already cryptographically
+%% secure PRNG. But, as stated previously,
+%% such a property does not mean that, whatever the
+%% key size, no attacker can predict the next bit
+%% knowing all the previously released ones.
+%% However, given a key size, it is possible to 
+%% measure in practice the minimum duration needed
+%% for an attacker to break a cryptographically
+%% secure PRNG, if we know the power of his/her
+%% machines. Such a concrete security evaluation 
+%% is related to the $(T,\varepsilon)-$security
+%% notion, which is recalled and evaluated in what 
+%% follows, for the sake of completeness.
 
-Let us recall that the running time of a probabilistic algorithm is defined to be the
-maximum of the expected number of steps needed to produce an output, maximized
-over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
-We are now able to define the notion of cryptographically secure PRNGs:
+%% Let us firstly recall that,
+%% \begin{definition}
+%% Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
+%% in time $T$. 
+%% Let $\varepsilon > 0$. 
+%% $\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
+%% generator $G$ if
 
-\begin{definition}
-A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
-\end{definition}
+%% \begin{flushleft}
+%% $\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
+%% \end{flushleft}
 
+%% \begin{flushright}
+%% $ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
+%% \end{flushright}
 
+%% \noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
+%% ``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
+%% corresponding set.
+%% \end{definition}
 
+%% Let us recall that the running time of a probabilistic algorithm is defined to be the
+%% maximum of the expected number of steps needed to produce an output, maximized
+%% over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
+%% We are now able to define the notion of cryptographically secure PRNGs:
 
+%% \begin{definition}
+%% A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
+%% \end{definition}
 
 
 
-Suppose now that the PRNG of Eq.~\eqref{equation Oplus} will work during 
-$M=100$ time units, and that during this period,
-an attacker can realize $10^{12}$ clock cycles.
-We thus wonder whether, during the PRNG's 
-lifetime, the attacker can distinguish this 
-sequence from a truly random one, with a probability
-greater than $\varepsilon = 0.2$.
-We consider that $N$ has 900 bits.
 
-Predicting the next generated bit knowing all the
-previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
-next bit in the BBS generator, which
-is cryptographically secure. More precisely, it
-is $(T,\varepsilon)-$secure: no 
-$(T,\varepsilon)-$distinguishing attack can be
-successfully realized on this PRNG, if~\cite{Fischlin}
-\begin{equation}
-T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
-\label{mesureConcrete}
-\end{equation}
-where $M$ is the length of the output ($M=100$ in
-our example), and $L(N)$ is equal to
-$$
-2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~  2))^\frac{2}{3}\right)
-$$
-is the number of clock cycles to factor a $N-$bit
-integer.
+
+
+
+%% Suppose now that the PRNG of Eq.~\eqref{equation Oplus} will work during 
+%% $M=100$ time units, and that during this period,
+%% an attacker can realize $10^{12}$ clock cycles.
+%% We thus wonder whether, during the PRNG's 
+%% lifetime, the attacker can distinguish this 
+%% sequence from a truly random one, with a probability
+%% greater than $\varepsilon = 0.2$.
+%% We consider that $N$ has 900 bits.
+
+%% Predicting the next generated bit knowing all the
+%% previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
+%% next bit in the BBS generator, which
+%% is cryptographically secure. More precisely, it
+%% is $(T,\varepsilon)-$secure: no 
+%% $(T,\varepsilon)-$distinguishing attack can be
+%% successfully realized on this PRNG, if~\cite{Fischlin}
+%% \begin{equation}
+%% T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
+%% \label{mesureConcrete}
+%% \end{equation}
+%% where $M$ is the length of the output ($M=100$ in
+%% our example), and $L(N)$ is equal to
+%% $$
+%% 2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~  2))^\frac{2}{3}\right)
+%% $$
+%% is the number of clock cycles to factor a $N-$bit
+%% integer.
 
 
 
 
-A direct numerical application shows that this attacker 
-cannot achieve its $(10^{12},0.2)$ distinguishing
-attack in that context.
+%% A direct numerical application shows that this attacker 
+%% cannot achieve its $(10^{12},0.2)$ distinguishing
+%% attack in that context.
 
 
 
@@ -1982,7 +2012,7 @@ on GPU can be useful in security context with the
 proposed parameters, or if it is only a very fast
 and statistically perfect generator on GPU, its
 $(T,\varepsilon)-$security must be determined, and
-a formulation similar to Eq.\eqref{mesureConcrete}
+a formulation similar to Annex~\ref{A-sec:Practicak evaluation} %.Eq.\eqref{mesureConcrete}
 must be established. Authors
 hope to achieve this difficult task in a future
 work.
