In most cases, these generators simply consist in iterating a chaotic function like
the logistic map~\cite{915396,915385} or the Arnold's one~\cite{5376454}\ldots
It thus remains to find optimal parameters in such functions so that attractors are
In most cases, these generators simply consist in iterating a chaotic function like
the logistic map~\cite{915396,915385} or the Arnold's one~\cite{5376454}\ldots
It thus remains to find optimal parameters in such functions so that attractors are
In order to check the quality of the produced outputs, it is usual to test the
PRNGs (Pseudo-Random Number Generators) with statistical batteries like
In order to check the quality of the produced outputs, it is usual to test the
PRNGs (Pseudo-Random Number Generators) with statistical batteries like
sensitiveness to the initial conditions (the well known ``butterfly effect''):
a continuous function $k$ defined on a metrical space is said
sensitiveness to the initial conditions (the well known ``butterfly effect''):
a continuous function $k$ defined on a metrical space is said
-\emph{strongly sensitive to the initial conditions} if for all point
-$x$ and all positive value $\epsilon$, it is possible to find another
-point $y$, as close as possible to $x$, and an integer $t$ such that the distance
+\emph{strongly sensitive to the initial conditions} if for each point
+$x$ and each positive value $\epsilon$, it is possible to find another
+point $y$ as close as possible to $x$, and an integer $t$ such that the distance
between the $t$-th iterates of $x$ and $y$, denoted by $k^t(x)$ and $k^t(y)$,
are larger than $\epsilon$. However, in his definition of chaos, Devaney~\cite{Devaney}
between the $t$-th iterates of $x$ and $y$, denoted by $k^t(x)$ and $k^t(y)$,
are larger than $\epsilon$. However, in his definition of chaos, Devaney~\cite{Devaney}
\emph{transitivity} and \emph{regularity}. Functions evoked above have
been studied according to these properties, and they have been proven as chaotic on $\R$.
But nothing guarantees that such properties are preserved when iterating the functions
\emph{transitivity} and \emph{regularity}. Functions evoked above have
been studied according to these properties, and they have been proven as chaotic on $\R$.
But nothing guarantees that such properties are preserved when iterating the functions
and sufficient that the Markov matrix associated to this graph is doubly stochastic,
in order to have a uniform distribution of the outputs. We have finally established
sufficient conditions to guarantee the first property of connectivity. Among the
and sufficient that the Markov matrix associated to this graph is doubly stochastic,
in order to have a uniform distribution of the outputs. We have finally established
sufficient conditions to guarantee the first property of connectivity. Among the
satisfy the second property too. In~\cite{chgw14oip}, we have proposed an algorithmic
method allowing to directly obtain a strongly connected iteration graph having a doubly
stochastic Markov matrix. The research work presented here generalizes this latter article
satisfy the second property too. In~\cite{chgw14oip}, we have proposed an algorithmic
method allowing to directly obtain a strongly connected iteration graph having a doubly
stochastic Markov matrix. The research work presented here generalizes this latter article
preliminaries, basic notations, and terminologies regarding asynchronous iterations.
Then, in Section~\ref{sec:proofOfChaos}, Devaney's definition of chaos is recalled
while the proofs of chaos of our most general PRNGs is provided. Section~\ref{sec:SCCfunc} shows how to generate functions and a number of iterations such that the iteration graph is strongly connected, making the
preliminaries, basic notations, and terminologies regarding asynchronous iterations.
Then, in Section~\ref{sec:proofOfChaos}, Devaney's definition of chaos is recalled
while the proofs of chaos of our most general PRNGs is provided. Section~\ref{sec:SCCfunc} shows how to generate functions and a number of iterations such that the iteration graph is strongly connected, making the
hypercube, while Section~\ref{sec:prng} establishes the link between the theoretical study and
pseudorandom number generation.
This research work ends by a conclusion section, where the contribution is summarized and
hypercube, while Section~\ref{sec:prng} establishes the link between the theoretical study and
pseudorandom number generation.
This research work ends by a conclusion section, where the contribution is summarized and
