X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/428720e875baf64678c06f2c75ae55185b18d81a..935ffe48008761ee21d78930849f51e32c62fcec:/intro.tex

diff --git a/intro.tex b/intro.tex
index f4ed222..569c493 100644
--- a/intro.tex
+++ b/intro.tex
@@ -4,20 +4,20 @@ chosen due to their unpredictable character and their sensitiveness to initial c
 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
@@ -38,7 +38,7 @@ asynchronous iterations are strongly connected. We then have proven that it is n
 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
@@ -215,6 +215,16 @@ theoretical properties but with a reduced mixing time.
 % 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