+\section{Basic Recalls}
+\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}
+
+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 notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
+
+
+Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : \mathcal{X} \rightarrow \mathcal{X}$.
+
+\begin{definition}
+$f$ is said to be \emph{topologically transitive} if, for any pair of open sets $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq \varnothing$.
+\end{definition}
+
+\begin{definition}
+An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$ if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$
+\end{definition}
+
+\begin{definition}
+$f$ is said to be \emph{regular} on $(\mathcal{X}, \tau)$ if the set of periodic points for $f$ is dense in $\mathcal{X}$: for any point $x$ in $\mathcal{X}$, any neighborhood of $x$ contains at least one periodic point (without necessarily the same period).
+\end{definition}
+
+
+\begin{definition}
+$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and topologically transitive.
+\end{definition}
+
+The chaos property is strongly linked to the notion of ``sensitivity'', defined on a metric space $(\mathcal{X},d)$ by:
+
+\begin{definition}
+\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
+if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that $d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
+
+$\delta$ is called the \emph{constant of sensitivity} of $f$.
+\end{definition}
+
+Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has the property of sensitive dependence on initial conditions (this property was formerly an element of the definition of chaos). To sum up, quoting Devaney in~\cite{Devaney}, a chaotic dynamical system ``is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or simplified into two subsystems which do not interact because of topological transitivity. And in the midst of this random behavior, we nevertheless have an element of regularity''. Fundamentally different behaviors are consequently possible and occur in an unpredictable way.
+
+
+
+\subsection{Chaotic iterations}
+\label{sec:chaotic iterations}
+
+
+Let us consider a \emph{system} with a finite number $\mathsf{N} \in
+\mathds{N}^*$ of elements (or \emph{cells}), so that each cell has a
+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 $\mathbb{S}.$
+
+\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
+\emph{chaotic iterations} are defined by $x^0\in
+\mathds{B}^{\mathsf{N}}$ and
+$$
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+\begin{array}{ll}
+ x_i^{n-1} & \text{ if }S^n\neq i \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.
+\end{array}\right.
+$$
+\end{definition}
+
+In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
+\textquotedblleft iterated\textquotedblright . Note that in a more
+general formulation, $S^n$ can be a subset of components and
+$\left(f(x^{n-1})\right)_{S^{n}}$ 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.
+
+
+Let us now recall how to define a suitable metric space where chaotic iterations are continuous. For further explanations, see, e.g., \cite{guyeux10}.
+
+Let $\delta $ be the \emph{discrete Boolean metric}, $\delta (x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
+\begin{equation*}
+\begin{array}{lrll}
+F_{f}: & \llbracket1;\mathsf{N}\rrbracket\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},%
+\end{array}%
+\end{equation*}%
+\noindent where + and . are the Boolean addition and product operations.
+Consider the phase space:
+\begin{equation*}
+\mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\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}
+\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:
+\begin{equation*}
+\left\{
+\begin{array}{l}
+X^0 \in \mathcal{X} \\
+X^{k+1}=G_{f}(X^k).%
+\end{array}%
+\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.
+
+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:
+\begin{equation*}
+d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
+\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})}, \\
+\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
+\sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
+\end{array}%
+\right.
+\end{equation*}
+
+
+This new distance has been introduced to satisfy the following requirements.
+\begin{itemize}
+\item When the number of different cells between two systems is increasing, then their distance should increase too.
+\item In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a while. Indeed, the two dynamical systems start with the same initial condition, use the same update function, and as strategies are the same for a while, then 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 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.
+
+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 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 introduced the notion of asynchronous iteration graph recalled bellow.
+
+Let $f$ be a map from $\mathds{B}^n$ 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 $i\in \llbracket1;n\rrbracket$,
+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
+strategy $s$ such that the parallel iteration of $G_f$ from the
+initial point $(s,x)$ reaches the point $x'$.
+
+We have finally proven in \cite{FCT11} 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)
+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 $G_f$ during implementations (due to the discrete nature of $f$). It is as if $\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ; n \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
+
+\section{Application to Pseudo-Randomness}
+
+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{scriptsize}
+\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)}
+\KwOut{a configuration $x$ ($n$ bits)}
+$x\leftarrow x^0$\;
+$k\leftarrow b + \textit{XORshift}(b+1)$\;
+\For{$i=0,\dots,k-1$}
+{
+$s\leftarrow{\textit{XORshift}(n)}$\;
+$x\leftarrow{F_f(s,x)}$\;
+}
+return $x$\;
+%\end{scriptsize}
+\caption{PRNG with chaotic functions}
+\label{CI Algorithm}
+\end{algorithm}
+
+\begin{algorithm}[h!]
+%\SetAlgoLined %%RAPH: cette ligne provoque une erreur chez moi
+\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 $y$\;
+\medskip
+\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 function $f$;
+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 returns 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.
+
+
+We have proven in \cite{FCT11} 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 as in the previous lemma.
+ 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}
+
+
+
+
+\section{Efficient prng based on chaotic iterations}
+
+In order to implement efficiently a PRNG based on chaotic iterations it is
+possible to improve previous works [ref]. One solution consists in considering
+that the strategy used contains all the bits for which the negation is
+achieved out. Then in order to apply the negation on these bits we can simply
+apply the xor operator between the current number and the strategy. In
+order to obtain the strategy we also use a classical PRNG.
+
+Here is an example with 16-bits numbers showing how the bitwise operations are
+applied. Suppose that $x$ and the strategy $S^i$ are defined in binary mode.
+Then the following table shows the result of $x$ xor $S^i$.
+$$
+\begin{array}{|cc|cccccccccccccccc|}
+\hline
+x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+\hline
+S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+\hline
+x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+\hline
+
+\hline
+ \end{array}
+$$
+
+%% \begin{figure}[htbp]
+%% \begin{center}
+%% \fbox{
+%% \begin{minipage}{14cm}
+%% unsigned int CIprng() \{\\
+%% static unsigned int x = 123123123;\\
+%% unsigned long t1 = xorshift();\\
+%% unsigned long t2 = xor128();\\
+%% unsigned long t3 = xorwow();\\
+%% x = x\textasciicircum (unsigned int)t1;\\
+%% x = x\textasciicircum (unsigned int)(t2$>>$32);\\
+%% x = x\textasciicircum (unsigned int)(t3$>>$32);\\
+%% x = x\textasciicircum (unsigned int)t2;\\
+%% x = x\textasciicircum (unsigned int)(t1$>>$32);\\
+%% x = x\textasciicircum (unsigned int)t3;\\
+%% return x;\\
+%% \}
+%% \end{minipage}
+%% }
+%% \end{center}
+%% \caption{sequential Chaotic Iteration PRNG}
+%% \label{algo:seqCIprng}
+%% \end{figure}
+
+
+
+\lstset{language=C,caption={C code of the sequential chaotic iterations based PRNG},label=algo:seqCIprng}
+\begin{lstlisting}
+unsigned int CIprng() {
+ static unsigned int x = 123123123;
+ unsigned long t1 = xorshift();
+ unsigned long t2 = xor128();
+ unsigned long t3 = xorwow();
+ x = x^(unsigned int)t1;
+ x = x^(unsigned int)(t2>>32);
+ x = x^(unsigned int)(t3>>32);
+ x = x^(unsigned int)t2;
+ x = x^(unsigned int)(t1>>32);
+ x = x^(unsigned int)t3;
+ return x;
+}
+\end{lstlisting}
+
+
+
+
+
+In listing~\ref{algo:seqCIprng} a sequential version of our chaotic iterations
+based PRNG is presented. The xor operator is represented by
+\textasciicircum. This function uses three classical 64-bits PRNG: the
+\texttt{xorshift}, the \texttt{xor128} and the \texttt{xorwow}. In the
+following, we call them xor-like PRNGSs. These three PRNGs are presented
+in~\cite{Marsaglia2003}. As each xor-like PRNG used works with 64-bits and as
+our PRNG works with 32-bits, the use of \texttt{(unsigned int)} selects the 32
+least significant bits whereas \texttt{(unsigned int)(t3$>>$32)} selects the 32
+most significants bits of the variable \texttt{t}. So to produce a random
+number realizes 6 xor operations with 6 32-bits numbers produced by 3 64-bits
+PRNG. This version successes the BigCrush of the TestU01 battery [P. L’ecuyer
+ and R. Simard. Testu01].
+
+\section{Efficient prng based on chaotic iterations on GPU}
+
+In order to benefit from computing power of GPU, a program needs to define
+independent blocks of threads which can be computed simultaneously. In general,
+the larger the number of threads is, the more local memory is used and the less
+branching instructions are used (if, while, ...), the better performance is
+obtained on GPU. So with algorithm \ref{algo:seqCIprng} presented in the
+previous section, it is possible to build a similar program which computes PRNG
+on GPU. In the CUDA [ref] environment, threads have a local identificator,
+called \texttt{ThreadIdx} relative to the block containing them.
+
+
+\subsection{Naive version for GPU}
+
+From the CPU version, it is possible to obtain a quite similar version for GPU.
+The principe consists in assigning the computation of a PRNG as in sequential to
+each thread of the GPU. Of course, it is essential that the three xor-like
+PRNGs used for our computation have different parameters. So we chose them
+randomly with another PRNG. As the initialisation is performed by the CPU, we
+have chosen to use the ISAAC PRNG [ref] to initalize all the parameters for the
+GPU version of our PRNG. The implementation of the three xor-like PRNGs is
+straightforward as soon as their parameters have been allocated in the GPU
+memory. Each xor-like PRNGs used works with an internal number $x$ which keeps
+the last generated random numbers. Other internal variables are also used by the
+xor-like PRNGs. More precisely, the implementation of the xor128, the xorshift
+and the xorwow respectively require 4, 5 and 6 unsigned long as internal
+variables.
+
+\begin{algorithm}
+
+\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like PRNGs in global memory\;
+NumThreads: Number of threads\;}
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadIdx is concerned by the computation} {
+ retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
+ \For{i=1 to n} {
+ compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
+ store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
+ }
+ store internal variables in InternalVarXorLikeArray[threadIdx]\;
+}
+
+\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
+\label{algo:gpu_kernel}
+\end{algorithm}
+
+Algorithm~\ref{algo:gpu_kernel} presents a naive implementation of PRNG using
+GPU. According to the available memory in the GPU and the number of threads
+used simultenaously, the number of random numbers that a thread can generate
+inside a kernel is limited, i.e. the variable \texttt{n} in
+algorithm~\ref{algo:gpu_kernel}. For example, if $100,000$ threads are used and
+if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}
+then the memory required to store internals variables of xor-like
+PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
+and random number of our PRNG is equals to $100,000\times ((4+5+6)\times
+2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb.
+
+All the tests performed to pass the BigCrush of TestU01 succeeded. Different
+number of threads, called \texttt{NumThreads} in our algorithm, have been tested
+upto $10$ millions.
