+Finally, we have established in \cite{bcgr11:ip} that,
+\begin{theorem}
+ Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
+ iteration graph, $\check{M}$ its adjacency
+ matrix and $M$
+ a $n\times n$ matrix defined by
+ $
+ M_{ij} = \frac{1}{n}\check{M}_{ij}$ %\textrm{
+ if $i \neq j$ and
+ $M_{ii} = 1 - \frac{1}{n} \sum\limits_{j=1, j\neq i}^n \check{M}_{ij}$ otherwise.
+
+ If $\Gamma(f)$ is strongly connected, then
+ the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows
+ a law that tends to the uniform distribution
+ if and only if $M$ is a double stochastic matrix.
+\end{theorem}
+
+
+These results of chaos and uniform distribution have led us to study the possibility of building a
+pseudorandom number generator (PRNG) based on the chaotic iterations.
+As $G_f$, defined on the domain $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}}
+\times \mathds{B}^\mathsf{N}$, is built 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 $f$). Indeed, it is as if
+$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ; \mathsf{N}
+\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).
+Let us finally remark that the vectorial negation satisfies the hypotheses of both theorems above.
+
+\section{Application to Pseudorandomness}
+\label{sec:pseudorandom}
+
+\subsection{A First Pseudorandom Number Generator}
+
+We have proposed in~\cite{bgw09:ip} a new family of generators that receives
+two PRNGs as inputs. These two generators are mixed with chaotic iterations,
+leading thus to a new PRNG that improves the statistical properties of each
+generator taken alone. Furthermore, our generator
+possesses various chaos properties that none of the generators used as input
+present.
+
+
+\begin{algorithm}[h!]
+\begin{small}
+\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
+($n$ bits)}
+\KwOut{a configuration $x$ ($n$ bits)}
+$x\leftarrow x^0$\;
+$k\leftarrow b + \textit{XORshift}(b)$\;
+\For{$i=0,\dots,k$}
+{
+$s\leftarrow{\textit{XORshift}(n)}$\;
+$x\leftarrow{F_f(s,x)}$\;
+}
+return $x$\;
+\end{small}
+\caption{PRNG with chaotic functions}
+\label{CI Algorithm}
+\end{algorithm}
+
+
+
+
+\begin{algorithm}[h!]
+\begin{small}
+\KwIn{the internal configuration $z$ (a 32-bit word)}
+\KwOut{$y$ (a 32-bit word)}
+$z\leftarrow{z\oplus{(z\ll13)}}$\;
+$z\leftarrow{z\oplus{(z\gg17)}}$\;
+$z\leftarrow{z\oplus{(z\ll5)}}$\;
+$y\leftarrow{z}$\;
+return $y$\;
+\end{small}
+\caption{An arbitrary round of \textit{XORshift} algorithm}
+\label{XORshift}
+\end{algorithm}
+
+
+
+
+
+This generator is synthesized in Algorithm~\ref{CI Algorithm}.
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques};
+an integer $b$, ensuring that the number of executed iterations is at least $b$
+and at most $2b+1$; and an initial configuration $x^0$.
+It returns the new generated configuration $x$. Internally, it embeds two
+\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that return integers
+uniformly distributed
+into $\llbracket 1 ; k \rrbracket$.
+\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia,
+which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
+with a bit shifted version of it. This PRNG, which has a period of
+$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
+in our PRNG to compute the strategy length and the strategy elements.
+
+This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.
+
+\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
+to a kind of merger of the two sequences used in Algorithm
+\ref{CI Algorithm}. When the updating function is the vectorial negation,
+this algorithm can be rewritten as follows:
+
+\begin{equation}
+\left\{
+\begin{array}{l}
+x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
+\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
+\end{array}
+\right.
+\label{equation Oplus}
+\end{equation}
+where $\oplus$ is for the bitwise exclusive or between two integers.
+This rewriting can be understood as follows. The $n-$th term $S^n$ of the
+sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents
+the list of cells to update in the state $x^n$ of the system (represented
+as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
+component of this state (a binary digit) changes if and only if the $k-$th
+digit in the binary decomposition of $S^n$ is 1.
+
+The single basic component presented in Eq.~\ref{equation Oplus} is of
+ordinary use as a good elementary brick in various PRNGs. It corresponds
+to the following discrete dynamical system in chaotic iterations:
+
+\begin{equation}
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+\begin{array}{ll}
+ x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+\end{array}\right.
+\label{eq:generalIC}
+\end{equation}
+where $f$ is the vectorial negation and $\forall n \in \mathds{N}$,
+$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
+$k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary
+decomposition of $S^n$ is 1. Such chaotic iterations are more general
+than the ones presented in Definition \ref{Def:chaotic iterations} because, instead of updating only one term at each iteration,
+we select a subset of components to change.
+
+
+Obviously, replacing Algorithm~\ref{CI Algorithm} by
+Equation~\ref{equation Oplus}, which is possible when the iteration function is
+the vectorial negation, leads to a speed improvement. However, proofs
+of chaos obtained in~\cite{bg10:ij} have been established
+only for chaotic iterations of the form presented in Definition
+\ref{Def:chaotic iterations}. The question is now to determine whether the
+use of more general chaotic iterations to generate pseudorandom numbers
+faster, does not deflate their topological chaos properties.
+
+\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+\label{deuxième def}
+Let us consider the discrete dynamical systems in chaotic iterations having
+the general form:
+
+\begin{equation}
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+\begin{array}{ll}
+ x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+\end{array}\right.
+\label{general CIs}
+\end{equation}
+
+In other words, at the $n^{th}$ iteration, only the cells whose id is
+contained into the set $S^{n}$ are iterated.
+
+Let us now rewrite these general chaotic iterations as usual discrete dynamical
+system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
+is required in order to study the topological behavior of the system.
+
+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:
+%%RAPH : j'ai coupé la dernière ligne en 2, c'est moche
+\begin{equation}
+\begin{array}{lrll}
+F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
+\longrightarrow & \mathds{B}^{\mathsf{N}} \\
+& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+\right.\\
+& & &\left.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 viré ce label qui existe déjà 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}%
+
+Once more, a shift function appears as a component of these general chaotic
+iterations.
+
+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 supprimé tous les sauts à 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
+$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
