+\subsection{A Metric Space for PRNG Iterations}
+
+% Define by $\mathcal{S}_X$ the set of sequences whose elements belong in $X \subset \mathds{N}, X \neq \varnothing$,
+% that is, $\mathcal{S}_X = \mathcal{X}^\mathds{N}$.
+% Let $\mathsf{N} \in \mathds{N}^\ast$, $f:\mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$, and
+% $\mathcal{P} \subset \mathds{N}^\ast$ a non empty and finite set of integers.
+
+% Any couple $(u,v) \in \mathcal{S}_{\llbracket 1, \mathsf{N} \rrbracket} \times \mathcal{S}_\mathcal{P}$ defines
+% a ``chaotic iterations based'' pseudorandom number generator, which is denoted by $\textit{CIPRNG}_f^2(u,v)$~\cite{wbg10:ip}. It is
+% defined as follows:
+% \begin{equation}
+% \label{CIPRNGver2}
+% \left\{
+% \begin{array}{l}
+% x^0 \in \mathds{B}^\mathsf{N}\\
+% \forall n \in \mathds{N}, \forall i \in \llbracket 1, \mathsf{N} \rrbracket, x_i^{n+1} = \left\{ \begin{array}{ll} f(x^n)_i & \text{if }~ i=u^n \\ x_i^n & \text{else} \end{array} \right.\\
+% \forall n \in \mathds{N}, y^n = x^{v^n} .
+% \end{array}
+% \right.
+% \end{equation}
+% The outputted sequence produced by this generator is $\left(y^n\right)_{n \in \mathds{N}}$.
+% Remark that, given a sequence $S \in \mathcal{S}_{\llbracket 1, \mathsf{N} \rrbracket}$ called a ``chaotic strategy'',
+% the following way to iterate:
+% $$x^0 \in \mathds{B}^\mathsf{N}, \forall n \in \mathds{N}, \forall i \in \llbracket 1, \mathsf{N} \rrbracket, x_i^{n+1} = \left\{ \begin{array}{ll} f(x^n)_i & \text{if }~ i=S^n \\ x_i^n & \text{else} \end{array} \right. ,$$
+% is referred in the discrete mathematics literature as ``chaotic iterations''~\cite{Robert} (a terminology which has
+% \emph{a priori} no relation with the mathematical theory of chaos recalled previously), which
+% explains the name provided to these categories of pseudorandom number generators.
+
+
+% The formerly proposed $\textit{CIPRNG}_f^1(u)$~\cite{bgw09:ip,guyeuxTaiwan10} is equal to \linebreak $\textit{CIPRNG}_f^2\left(u,\left(1\right)_{n\in \mathds{N}}\right)$, where $\left(1\right)_{n\in \mathds{N}}$ is the sequence that is
+% uniformly equal to 1.
+% It has been proven as chaotic for the vectorial Boolean negation $f_0:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N}$,
+% $(x_1, \hdots , x_\mathsf{N}) \longmapsto (\overline{x_1}, \hdots, \overline{x_\mathsf{N}})$ in \cite{bgw09:ip}
+% and for a larger set of well-chosen iteration functions in~\cite{bcgr11:ip} but,
+% as only one bit is modified at each iteration, this generator is not able to pass any reasonable statistical tests.
+
+% The $\textit{XOR~CIPRNG}(S)$, for its part~\cite{DBLP:journals/corr/abs-1112-5239}, is defined as follows: $x^0 \in \mathds{B}^\mathsf{N}$, and $\forall n \in \mathds{N}, x^{n+1} = x^n \oplus S^n$
+% where $S \in \mathcal{S}_{\llbracket 1, \mathsf{N} \rrbracket}$ and $\oplus$ stands for the bitwise \emph{exclusive or} (xor) operation
+% between the binary decomposition of $x^n$ and $S^n$. This is indeed a $CIPRNG_{f_0}^2 (u,v)$ generator:
+% %, for
+% %$u,v \in \mathcal{S}_{\llbracket 1, \mathsf{N} \rrbracket}$:
+% for any given $S \in \mathcal{S}_{\llbracket 1, \mathsf{N} \rrbracket}$, $v^n$ is the number
+% of 1's in the binary decomposition of $S^n$ while $u^{v^n}, u^{v^n+1}, \hdots , u^{v^{n+1}-1}$
+% are the positions of these ones.
+% The $\textit{XOR~CIPRNG}$ has been proven chaotic and it is able to pass all the most stringent statistical
+% batteries of tests~\cite{DBLP:journals/corr/abs-1112-5239}, namely: DieHARD~\cite{Marsaglia1996}, NIST~\cite{Nist10}, and TestU01~\cite{LEcuyerS07},
+% which encompasses the two former ones. Furthermore, the output sequence is cryptographically secure
+% when $S$ is cryptographically secure~\cite{DBLP:journals/corr/abs-1112-5239}.
+% We are then left to prove $\textit{CIPRNG}_f^2(u,v)$ is chaotic.
+
+% \subsection{The $\textit{CIPRNG}_f^2(u,v)$ is chaotic for well-chosen $f$}\label{sec:wellchosen}
+
+% \subsection{The generator as a discrete dynamical system}
+
+
+% This algorithm may be seen as $\mathsf{p}$ functional composition of $F_f$.
+% We thus introduce the function
+% $F_{f,\mathsf{p}} : \mathds{B}^\mathsf{N} \times \llbracket 1, \mathsf{N} \rrbracket^\mathsf{p} \rightarrow \mathds{B}^\mathsf{N}$ defined by
+
+% $$
+% F_{f,\mathsf{p}} (x,(u^0, u^1, \hdots, u^{\mathsf{p}-1})) \mapsto
+% F_f(\hdots (F_f(F_f(x,u^0), u^1), \hdots), u^{\mathsf{p}-1}).
+% $$
+
+
+
+
+Let us first introduce $\mathcal{P} \subset \mathds{N}$ a finite nonempty
+set having the cardinality $\mathsf{p} \in \mathds{N}^\ast$.
+Intuitively, this is the set of authorized numbers of iterations.
+Denote by $p_1, p_2, \hdots, p_\mathsf{p}$ the ordered elements of $\mathcal{P}$: $\mathcal{P} = \{ p_1, p_2, \hdots, p_\mathsf{p}\}$
+and $p_1< p_2< \hdots < p_\mathsf{p}$. In our algorithm,
+$\mathsf{p}$ is 1 and $p_1$ is $b$.
