\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
-\title{Efficient generation of pseudo random numbers based on chaotic iterations
+\title{Efficient Generation of Pseudo-Random Bumbers based on Chaotic Iterations
on GPU}
\begin{document}
\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}
+\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
+is for the $k^{th}$ composition of a function $f$. Finally, the following
notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
\end{definition}
-\begin{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
topologically transitive.
\end{definition}
-\subsection{Chaotic iterations}
+\subsection{Chaotic Iterations}
\label{sec:chaotic iterations}
\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 $S\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ be a strategy. The so-called
+a function and $S\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ be a ``strategy''. The so-called
\emph{chaotic iterations} are defined by $x^0\in
\mathds{B}^{\mathsf{N}}$ and
\begin{equation}
$\left(f(x^{k})\right)_{S^{n}}$, 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.
+priori} no link with the mathematical theory of chaos, presented above.
Let us now recall how to define a suitable metric space where chaotic iterations
(S^{n})_{n\in \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow (S^{n+1})_{n\in
\mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ and $i$ is the \emph{initial function}
$i:(S^{n})_{n\in \mathds{N}} \in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow S^{0}\in \llbracket
-1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in
-(\ref{sec:chaotic iterations}) can be described by the following iterations:
+1;\mathsf{N}\rrbracket$. Then the chaotic iterations proposed in
+Definition \ref{Def:chaotic iterations} can be described by the following iterations:
\begin{equation}
\left\{
\begin{array}{l}
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:
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.
+The impact of this choice for a distance will be investigate at the end of the document.
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
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ 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
+Boolean negation $f(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \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
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ 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 $i\in \llbracket1;n\rrbracket$,
+$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and
+$i\in \llbracket1;\mathsf{N}\rrbracket$,
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
\begin{theorem}
\label{Th:Caractérisation des IC chaotiques}
-Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic (according to Devaney)
+Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ 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 $G_f$
+As $G_f$, defined on the domain $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}}
+\times \mathds{B}^\mathsf{N}$, is build 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 $f$). It is as if
-$\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ; n
+$\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).
\section{Application to Pseudo-Randomness}
if and only if $M$ is a double stochastic matrix.
\end{theorem}
+This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.
-
-\subsection{Improving the speed of the former generator}
+\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
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 pseudo-random numbers more
-fastly, does not deflate their topological chaos properties.
+use of more general chaotic iterations to generate pseudo-random numbers
+faster, does not deflate their topological chaos properties.
-\subsection{Proofs of chaos of the general formulation of the chaotic iterations}
+\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
Let us consider the discrete dynamical systems in chaotic iterations having
the general form:
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 introduced.
+$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
Let us introduce:
\begin{equation}
d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
\end{proof}
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,
+
+\begin{theorem}
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.
+\end{theorem}
+
+Let us firstly prove the following lemma.
+
+\begin{lemma}[Strong transitivity]
+\label{strongTrans}
+ For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can
+find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+\end{lemma}
+
+\begin{proof}
+ Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$.
+Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$,
+are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define
+$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+the form $(S',E')$ where $E'=E$ and $S'$ starts with
+$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+\begin{itemize}
+ \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+ \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+\end{itemize}
+Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$,
+where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+claimed in the lemma.
+\end{proof}
+
+We can now prove the Theorem~\ref{t:chaos des general}...
+
+\begin{proof}[Theorem~\ref{t:chaos des general}]
+Firstly, strong transitivity implies transitivity.
+
+Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+prove that $G_f$ is regular, it is sufficient to prove that
+there exists a strategy $\tilde S$ such that the distance between
+$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+$(\tilde S,E)$ is a periodic point.
+
+Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$
+and $t_2\in\mathds{N}$ such
+that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+of $S$ and the first $t_2$ terms of $S'$: $$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
+$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
+point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
+have $d((S,E),(\tilde S,E))<\epsilon$.
+\end{proof}
