-% Dans cet article, nous avons montré qu'une fonction $G_f$ est chaotique si et
-% seulement la fonction booléenne $f$ a un graphe d'itérations chaotiques
-% fortement connexe. L'originalité majeure repose sur le type d'itérations
-% considéré, qui n'est pas limité à la mise à jour d'un seul élément par
-% itération, mais qui est étendu à la mise à jour simultanée de plusieurs éléments
-% du système à chaque itération. De plus, il a été prouvé que la sortie d'une
-% telle fonction suit une loi de distribution uniforme si et seulement si la
-% chaîne de Markov induite peut se représenter à l'aide d'une matrice doublement
-% stochastique. Enfin, un algorithme permettant d'engendrer des fonctions qui
-% vérifient ces deux contraintes a été présenté et évalué. Ces fonctions ont été
-% ensuite appliquées avec succès à la génération de nombres pseudo-aléatoires.
-% Les expériences sur une batterie de tests éprouvée ont pu confirmer la
-% pertinence de l'approche théorique.
-%
+This work has assumed a Boolean map $f$ which is embedded into
+a discrete-time dynamical system $G_f$.
+This one is supposed to be iterated a fixed number
+$p_1$ or $p_2$,\ldots, or $p_{\mathds{p}}$ of
+times before its output is considered.
+This work has first shown that iterations of
+$G_f$ are chaotic if and only if its iteration graph $\Gamma_{\mathcal{P}}(f)$
+is strongly connected where $\mathcal{P}$ is $\{p_1, \ldots, p_{\mathds{p}}\}$.
+Any PRNG, which iterates $G_f$ as above
+satisfies in some cases the property of chaos.
-In this article, we have proven that the most general chaotic iterations based PRNG
-satisfy the property of chaos as defined by Devaney. We then have shown how to generate
-such functions together with the number of iterations, leading to strongly connected
-iteration graphs and thus to chaos for the associated pseudorandom number generators.
-By removing some paths in the hypercube, we then have provided examples of such graphs
-that lead to chaos, while relating these graphs to the PRNG problem under consideration.
+We then have shown that a previously presented approach can be directly
+applied here to generate function $f$ with strongly connected
+$\Gamma_{\mathcal{P}}(f)$.
+The iterated map inside the generator is built by first removing from a
+$\mathsf{N}$-cube an Hamiltonian path and next
+adding a self loop to each vertex.
+The PRNG can thus be seen as a random walks of length in $\mathsf{P}$
+into $\mathsf{N}$ this new cube.
+We furthermore have exhibit a bound on the number of iterations
+that are sufficient to obtain a uniform distribution of the output.
+Finally, experiments through the NIST battery have shown that
+the statistical properties are almost established for
+$\mathsf{N} = 4, 5, 6, 7, 8$.
-In future work, we intend to understand the link between succeeded or failed statistical tests
-and the properties of chaos for the associated asynchronous iterations. By doing so,
-relations between desired statistically unbiased behaviors and topological properties will be
-understood, leading to better choices in iteration functions. Conditions allowing the
-reduction of the mixing time will be investigated too, while other modifications of the hypercube
-will be regarded, in order to enlarge the set of known chaotic and random asynchronous
-iterations.
-
-%
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+In future work, we intend to understand the link between
+statistical tests and the properties of chaos for
+the associated iterations.
+By doing so, relations between desired statistically unbiased behaviors and
+topological properties will be understood, leading to better choices
+in iteration functions.
+Conditions allowing the reduction of the stopping-time will be
+investigated too, while other modifications of the hypercube will
+be regarded in order to enlarge the set of known chaotic
+and random iterations.
