X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/8657185ec89ecf42d39c56a9a57ec20e61e20299..a6692cd736d836866212aae44ca8d787b63b1d01:/prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 5a3324b..2a27439 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 numbers based on chaotic iterations
+\title{Efficient Generation of Pseudo-Random Bumbers based on Chaotic Iterations
 on GPU}
 \begin{document}
 
@@ -59,11 +59,11 @@ Interet de générer des nombres alea sur GPU
 \label{section:BASIC RECALLS}
 This section is devoted to basic definitions and terminologies in the fields of
 topological chaos and chaotic iterations.
-\subsection{Devaney's chaotic dynamical systems}
+\subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
-denotes the $k^{th}$ composition of a function $f$. Finally, the following
+is for the $k^{th}$ composition of a function $f$. Finally, the following
 notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
 
 
@@ -89,7 +89,7 @@ necessarily the same period).
 \end{definition}
 
 
-\begin{definition}
+\begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
 $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
 topologically transitive.
 \end{definition}
@@ -119,7 +119,7 @@ possible and occur in an unpredictable way.
 
 
 
-\subsection{Chaotic iterations}
+\subsection{Chaotic Iterations}
 \label{sec:chaotic iterations}
 
 
@@ -129,13 +129,13 @@ Boolean  \emph{state}. Having $\mathsf{N}$ Boolean values for these
  cells  leads to the definition of a particular \emph{state  of the
 system}. A sequence which  elements belong to $\llbracket 1;\mathsf{N}
 \rrbracket $ is called a \emph{strategy}. The set of all strategies is
-denoted by $\mathbb{S}.$
+denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$
 
 \begin{definition}
 \label{Def:chaotic iterations}
 The      set       $\mathds{B}$      denoting      $\{0,1\}$,      let
 $f:\mathds{B}^{\mathsf{N}}\longrightarrow  \mathds{B}^{\mathsf{N}}$ be
-a  function  and  $S\in  \mathbb{S}$  be  a  strategy.  The  so-called
+a  function  and  $S\in  \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$  be  a  ``strategy''.  The  so-called
 \emph{chaotic      iterations}     are     defined      by     $x^0\in
 \mathds{B}^{\mathsf{N}}$ and
 \begin{equation}
@@ -155,7 +155,7 @@ $\left(f(x^{n-1})\right)_{S^{n}}$      can     be      replaced     by
 $\left(f(x^{k})\right)_{S^{n}}$, where  $k<n$, describing for example,
 delays  transmission~\cite{Robert1986,guyeux10}.  Finally,  let us  remark that
 the term  ``chaotic'', in  the name of  these iterations,  has \emph{a
-priori} no link with the mathematical theory of chaos, recalled above.
+priori} no link with the mathematical theory of chaos, presented above.
 
 
 Let us now recall how to define a suitable metric space where chaotic iterations
@@ -182,11 +182,11 @@ Consider the phase space:
 G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
 \end{equation}
 \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
-(S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in
-\mathds{N}}\in \mathbb{S}$ and $i$ is the \emph{initial function} 
-$i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket
-1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in
-(\ref{sec:chaotic iterations}) can be described by the following iterations:
+(S^{n})_{n\in \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow (S^{n+1})_{n\in
+\mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ and $i$ is the \emph{initial function} 
+$i:(S^{n})_{n\in \mathds{N}} \in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow S^{0}\in \llbracket
+1;\mathsf{N}\rrbracket$. Then the chaotic iterations proposed in
+Definition \ref{Def:chaotic iterations} can be described by the following iterations:
 \begin{equation}
 \left\{
 \begin{array}{l}
@@ -200,7 +200,6 @@ With this formulation, a shift function appears as a component of chaotic
 iterations. The shift function is a famous example of a chaotic
 map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as
 chaotic. 
-
 To study this claim, a new distance between two points $X = (S,E), Y =
 (\check{S},\check{E})\in
 \mathcal{X}$ has been introduced in \cite{guyeux10} as follows:
@@ -233,27 +232,29 @@ components that are updated are the same too.
 \end{itemize}
 The distance presented above follows these recommendations. Indeed, if the floor
 value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$
-differ in $n$ cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a
+differ in $n$ cells ($d_e$ is indeed the Hamming distance). In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a
 measure of the differences between strategies $S$ and $\check{S}$. More
 precisely, this floating part is less than $10^{-k}$ if and only if the first
 $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is
 nonzero, then the $k^{th}$ terms of the two strategies are different.
+The impact of this choice for a distance will be investigate at the end of the document.
 
 Finally, it has been established in \cite{guyeux10} that,
 
 \begin{proposition}
-Let $f$ be a map from $\mathds{B}^n$ to itself. Then $G_{f}$ is continuous in
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. Then $G_{f}$ is continuous in
 the metric space $(\mathcal{X},d)$.
 \end{proposition}
 
 The chaotic property of $G_f$ has been firstly established for the vectorial
-Boolean negation \cite{guyeux10}. To obtain a characterization, we have secondly
+Boolean negation $f(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
 introduced the notion of asynchronous iteration graph recalled bellow.
 
-Let $f$ be a map from $\mathds{B}^n$ to itself. The
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
 {\emph{asynchronous iteration graph}} associated with $f$ is the
 directed graph $\Gamma(f)$ defined by: the set of vertices is
-$\mathds{B}^n$; for all $x\in\mathds{B}^n$ and $i\in \llbracket1;n\rrbracket$,
+$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and 
+$i\in \llbracket1;\mathsf{N}\rrbracket$,
 the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. 
 The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
@@ -265,17 +266,17 @@ We have finally proven in \cite{bcgr11:ip} that,
 
 \begin{theorem}
 \label{Th:Caractérisation   des   IC   chaotiques}  
-Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic  (according to  Devaney) 
+Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic  (according to  Devaney) 
 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
 This result of chaos has lead us to study the possibility to build a
 pseudo-random number generator (PRNG) based on the chaotic iterations. 
-As $G_f$, defined on the domain   $\llbracket 1 ;  n \rrbracket^{\mathds{N}} 
-\times \mathds{B}^n$, is build from Boolean networks $f : \mathds{B}^n
-\rightarrow \mathds{B}^n$, we can preserve the theoretical properties on $G_f$
+As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathds{N}} 
+\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
+\rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
 during implementations (due to the discrete nature of $f$). It is as if
-$\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ;  n
+$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
 \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
 
 \section{Application to Pseudo-Randomness}
@@ -350,9 +351,9 @@ We have proven in \cite{bcgr11:ip} that,
   if and only if $M$ is a double stochastic matrix.
 \end{theorem} 
 
+This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.
 
-
-\subsection{Improving the speed of the former generator}
+\subsection{Improving the Speed of the Former Generator}
 
 Instead of updating only one cell at each iteration, we can try to choose a
 subset of components and to update them together. Such an attempt leads
@@ -388,6 +389,7 @@ 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
@@ -404,10 +406,10 @@ the vectorial negation, leads to a speed improvement. However, proofs
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
 \ref{Def:chaotic iterations}. The question is now to determine whether the
-use of more general chaotic iterations to generate pseudo-random numbers more 
-fastly, does not deflate their topological chaos properties.
+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}
+\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
 
 Let us consider the discrete dynamical systems in chaotic iterations having 
 the general form:
@@ -419,6 +421,7 @@ the general form:
   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
@@ -431,7 +434,7 @@ is required in order to study the topological behavior of the system.
 Let us introduce the following function:
 \begin{equation}
 \begin{array}{cccc}
- \delta: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
+ \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}
@@ -440,16 +443,17 @@ where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $
 Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
 \begin{equation}
 \begin{array}{lrll}
-F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
+F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
 \longrightarrow & \mathds{B}^{\mathsf{N}} \\
-& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
-(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+& (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}%
-\noindent where + and . are the Boolean addition and product operations.
+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} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
+\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 $\mathcal{X}$:
@@ -457,11 +461,11 @@ Consider the phase space:
 G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
 \end{equation}
 \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
-(S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in
-\mathds{N}}\in \mathbb{S}$ and $i$ is the \emph{initial function} 
-$i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket
-1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in
-(\ref{sec:chaotic iterations}) can be described by the following iterations:
+(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}
@@ -471,28 +475,178 @@ X^{k+1}=G_{f}(X^k).%
 \right.
 \end{equation}%
 
-With this formulation, a shift function appears as a component of chaotic
-iterations. The shift function is a famous example of a chaotic
-map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as
-chaotic. 
+Another time, a shift function appears as a component of these general chaotic 
+iterations. 
 
-To study this claim, a new distance between two points $X = (S,E), Y =
-(\check{S},\check{E})\in
-\mathcal{X}$ has been introduced in \cite{guyeux10} as follows:
+To study the Devaney's chaos property, a distance between two points 
+$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
+Let us introduce:
 \begin{equation}
 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
+\label{nouveau d}
 \end{equation}
 \noindent where
 \begin{equation}
 \left\{
 \begin{array}{lll}
 \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
-}\delta (E_{k},\check{E}_{k})}, \\
+}\delta (E_{k},\check{E}_{k})}\textrm{ is another time the Hamming distance}, \\
 \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
-\sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
+\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 $d$ will be a distance as sum of two distances.
+ \begin{itemize}
+\item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
+$d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
+$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
+ \item $d_s$ is symmetric 
+($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
+of the symmetric difference. 
+\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, 
+and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, 
+we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
+inequality is obtained.
+ \end{itemize}
+\end{proof}
+
+
+Before being able to study the topological behavior of the general 
+chaotic iterations, we must firstly establish that:
+
+\begin{proposition}
+ For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 
+$\left( \mathcal{X},d\right)$.
+\end{proposition}
+
+
+\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 distance
+between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
+strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
+\medskip
+\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
+\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
+\begin{equation*}
+\exists n_{2}\in \mathds{N},\forall n\geqslant
+n_{2},d_{s}(S^n,S)<10^{-(k+2)},
+\end{equation*}%
+thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
+\end{itemize}
+\noindent As a consequence, the $k+1$ first entries of the strategies of $%
+G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
+the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
+10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline
+In conclusion,
+$$
+\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
+,\forall n\geqslant N_{0},
+ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+\leqslant \varepsilon .
+$$
+$G_{f}$ is consequently continuous.
+\end{proof}
+
+
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,
+
+\begin{theorem}
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.
+\end{theorem}
+
+Let us firstly prove the following lemma.
+
+\begin{lemma}[Strong transitivity]
+\label{strongTrans}
+ For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
+find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+\end{lemma}
+
+\begin{proof}
+ Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
+Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
+are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
+$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+the form $(S',E')$ where $E'=E$ and $S'$ starts with 
+$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+\begin{itemize}
+ \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+ \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+\end{itemize}
+Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
+where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+claimed in the lemma.
+\end{proof}
+
+We can now prove the Theorem~\ref{t:chaos des general}...
+
+\begin{proof}[Theorem~\ref{t:chaos des general}]
+Firstly, strong transitivity implies transitivity.
+
+Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+prove that $G_f$ is regular, it is sufficient to prove that
+there exists a strategy $\tilde S$ such that the distance between
+$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+$(\tilde S,E)$ is a periodic point.
+
+Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
+and $t_2\in\mathds{N}$ such
+that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+of $S$ and the first $t_2$ terms of $S'$: $$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
+$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
+point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
+have $d((S,E),(\tilde S,E))<\epsilon$.
+\end{proof}
+
 
 
 \section{Efficient PRNG based on Chaotic Iterations}
@@ -657,7 +811,7 @@ 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
@@ -702,7 +856,28 @@ version}
 \label{algo:gpu_kernel2}
 \end{algorithm}
 
-
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having 
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, four iterations of the general chaotic
+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}