\author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe
Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
-\maketitle
+\IEEEcompsoctitleabstractindextext{
\begin{abstract}
In this paper we present a new pseudorandom number generator (PRNG) on
graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations. It
\end{abstract}
+}
+
+\maketitle
+
+\IEEEdisplaynotcompsoctitleabstractindextext
+\IEEEpeerreviewmaketitle
+
\section{Introduction}
\label{section:BASIC RECALLS}
This section is devoted to basic definitions and terminologies in the fields of
-topological chaos and chaotic iterations.
+topological chaos and chaotic iterations. We assume the reader is familiar
+with basic notions on topology (see for instance~\cite{Devaney}).
+
+
\subsection{Devaney's Chaotic Dynamical Systems}
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
\mathcal{X} \rightarrow \mathcal{X}$.
\begin{definition}
-$f$ is said to be \emph{topologically transitive} if, for any pair of open sets
+The function $f$ 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}[Devaney's formulation of chaos~\cite{Devaney}]
-$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
+The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
topologically transitive.
\end{definition}
on a metric space $(\mathcal{X},d)$ by:
\begin{definition}
-\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
+\label{sensitivity} The function $f$ 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 $f$.
+The constant $\delta$ 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
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:
-%%RAPH : ici j'ai coupé la dernière ligne en 2, c'est moche mais bon
-\begin{equation}
+(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function
+$F_{f}: \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}}
+\longrightarrow \mathds{B}^{\mathsf{N}}$
+\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)+ \right.\\
-& & & \left. f(E)_{k}.\overline{\delta
-(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+& (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}%
+\end{equation*}%
\noindent where + and . are the Boolean addition and product operations.
Consider the phase space:
\begin{equation}
\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:
+the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in
+\llbracket1;\mathsf{N}\rrbracket $,
\begin{equation}
-\forall n\in \mathds{N}^{\ast }, \forall i\in
-\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+ 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.
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} $, define the function:
-%%RAPH : j'ai coupé la dernière ligne en 2, c'est moche
-\begin{equation}
-\begin{array}{lrll}
-F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
-\longrightarrow & \mathds{B}^{\mathsf{N}} \\
-& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+\right.\\
-& & &\left.f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+$F_{f}: \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}}
+\longrightarrow \mathds{B}^{\mathsf{N}}$
+\begin{equation*}
+\begin{array}{rll}
+ (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}%
+\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:
\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}
+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
\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
+10^{-(k+1)}\leqslant \varepsilon $.
+
In conclusion,
%%RAPH : ici j'ai rajouté une ligne
-\begin{flushleft}$$
-\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
-,\forall n\geqslant N_{0},$$
-$$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+$
+\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+,$ $\forall n\geqslant N_{0},$
+$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
\leqslant \varepsilon .
-$$
-\end{flushleft}
+$
$G_{f}$ is consequently continuous.
\end{proof}
claimed in the lemma.
\end{proof}
-We can now prove Theorem~\ref{t:chaos des general}...
+We can now prove the Theorem~\ref{t:chaos des general}.
\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.
by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
one has
+$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and,
+therefore,
\begin{equation}\label{PCH-2}
-\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1].
+\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1].
\end{equation}
Now, using (\ref{PCH-1}) again, one has for every $x$,
\end{equation}
where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$,
thus
-\begin{equation}\label{PCH-3}
+\begin{equation}%\label{PCH-3} %%RAPH : j'ai viré ce label qui existe déjà, il est 3 ligne avant
D^\prime(H(x))=D(yx),
\end{equation}
where $y$ is randomly generated.
