% pertinence de l'approche théorique.
%
-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
+In this article, we have proven that the most general chaotic iterations based PRNG, which embeds
+an iteration function, satisfies in some cases the property of chaos
+as defined by Devaney. We then have shown how to generate such functions together
+with the related 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 linking these graphs to the PRNG problem under consideration.
+
+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.
-% The next section focus 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.
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