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
-avoided, guaranteeing by doing so that generated numbers follow a uniform distribution.
+avoided, hoping by doing so that the generated numbers follow a uniform distribution.
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
-the so-called DieHARD~\cite{Marsaglia1996}, NIST~\cite{Nist10}, or TestU01~\cite{LEcuyerS07}.
+the so-called DieHARD~\cite{Marsaglia1996}, NIST~\cite{Nist10}, or TestU01~\cite{LEcuyerS07} ones.
-In its general understanding, the chaos notion is often reduced to the strong
+In its general understanding, chaos notion is often reduced to the strong
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}
-impose to the chaotic function two other properties called
+imposes to the chaotic function two other properties called
\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
-generated functions, we thus considered for further investigations only the one that
+generated functions, we thus have considered for further investigations only the one that
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
% sur une batterie de tests.
+The remainder of this article is organized as follows. The next section is devoted to
+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
+PRNG chaotic. The next section focuses on examples of such graphs obtained by modifying 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
+intended future work is outlined.
+
% Le reste de ce document est organisé comme suit.
% La section~\ref{section:chaos} présente ce qu'est un système dynamique discret booléen itérant une fonction $f$.
% La chaoticité de la fonction engendrée $G_f$ est caractérisée en
