X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/0ed548dcbbb0d4580a211e1c8de4e76f4c4208ad..935ffe48008761ee21d78930849f51e32c62fcec:/conclusion.tex?ds=inline

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 % 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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