+
+
+
+
+
+
+\subsection{Chaotic iterations}\index{chaotic iterations}
+\label{subsection:Chaotic iterations}
+
+Let us now introduce an example of a dynamical systems family that has
+the potentiality to become chaotic, depending on the choice of the iteration
+function. This family is the basis of the PRNGs we will
+develop during this chapter.
+
+\begin{definition}
+\label{Chaotic iterations}
+The set $\mathds{B}$ denoting $\{0,1\}$, let $f:\mathds{B}^{\mathsf{N}%
+}\longrightarrow \mathds{B}^{\mathsf{N}}$ be an ``iteration'' function and $S$ be a
+sequence of subsets of $\llbracket 1, \mathsf{N}\rrbracket$ called a chaotic strategy. Then, the so-called \emph{chaotic iterations} are defined by~\cite{Robert1986}:
+
+\begin{equation}
+\label{eq:generalIC}
+\left\{\begin{array}{l}
+x^0\in \mathds{B}^{\mathsf{N}}, \\
+\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 S^n \\
+f(x^{n-1})_{i} & \text{if}~ i \in S^n.
+\end{array}
+\right.
+\end{array}
+\right.
+\end{equation}
+\end{definition}
+
+In other words, at the $n^{th}$ iteration, only the cells of $S^{n}$ are
+``iterated''.
+Chaotic iterations generate a set of vectors;
+they are defined by an initial state $x^{0}$, an iteration function $f$, and a chaotic strategy $S$~\cite{bg10:ij}.
+These ``chaotic iterations'' can behave chaotically as defined by Devaney,
+depending on the choice of $f$~\cite{bg10:ij}. For instance,
+chaos is obtained when $f$ is the vectorial negation.
+Note that, with this example of function, chaotic iterations
+defined above can be rewritten as:
+\begin{equation}
+\label{equation Oplus}
+x^0 \in \llbracket 0,2^\mathsf{N}-1\rrbracket,~\mathcal{S}^n \in \mathcal{P}\left(\llbracket 1,2^\mathsf{N}-1\rrbracket\right)^\mathds{N},~~ x^{n+1}=x^n \oplus \mathcal{S}^n,
+\end{equation}
+where $\mathcal{P}(X)$ stands for the set of subsets of $X$, whereas
+$a\oplus b$ is the bitwise exclusive or operation between the binary
+representation of the integers $a$ and $b$. Note that the term
+ $\mathcal{S}^n$ is directly and obviously linked to the $S^n$ of
+Eq.\ref{eq:generalIC}. As recalled above, such an iterative sequence
+satisfies the Devaney's definition of chaos.
+
