\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
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.
\end{array}\right.
+\label{eq:generalIC}
\end{equation}
where $f$ is the vectorial negation and $\forall n \in \mathds{N}$,
$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
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}),
We can now prove the Theorem~\ref{t:chaos des general}...
\begin{proof}[Theorem~\ref{t:chaos des general}]
- On the one hand, strong transitivity implies transitivity. On the other hand,
-the regularity is exactly Lemma~\ref{strongTrans} with $Y=X$. As the sensitivity
-to the initial condition is implied by these two properties, we thus have
-the theorem.
+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}
As GPU cards using CUDA have shared memory between threads of the same block, it
is possible to use this feature in order to simplify the previous algorithm,
-i.e. using less than 3 xor-like PRNGs. The solution consists in computing only
+i.e., using less than 3 xor-like PRNGs. The solution consists in computing only
one xor-like PRNG by thread, saving it into shared memory and using the results
of some other threads in the same block of threads. In order to define which
thread uses the result of which other one, we can use a permutation array which
\label{algo:gpu_kernel2}
\end{algorithm}
-
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, four iterations of the general chaotic
+iterations are realized between two stored values of the PRNG.
+To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+we must guarantee that this dynamical system iterates on the space
+$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
+The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+To prevent from any flaws of chaotic properties, we must check that each right
+term, corresponding to terms of the strategies, can possibly be equal to any
+integer of $\llbracket 1, \mathsf{N} \rrbracket$.
+
+Such a result is obvious for the two first lines, as for the xor-like(), all the
+integers belonging into its interval of definition can occur at each iteration.
+It can be easily stated for the two last lines by an immediate mathematical
+induction.
+
+Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
+chaotic iterations presented previously, and for this reason, it satisfies the
+Devaney's formulation of a chaotic behavior.
\section{Experiments}
