typos

[rairo15.git] / conclusion.tex

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 % pertinence de l'approche théorique.
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 % pertinence de l'approche théorique.
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-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. 
 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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