From: couchot Date: Thu, 30 Jun 2016 12:46:24 +0000 (+0200) Subject: modif conclusion X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/commitdiff_plain/789817c68132962fe415afd50e1105d04fabea47?ds=sidebyside modif conclusion --- diff --git a/conclusion.tex b/conclusion.tex index 2705043..0de1fc3 100644 --- a/conclusion.tex +++ b/conclusion.tex @@ -6,22 +6,35 @@ 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. +It can be deduced that in such a situation a PRNG, which iterates $G_f$ +satisfies the property of chaos and can be used in simulating chaos +phenomenon. 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 +$\mathsf{N}$-cube a balanced Hamiltonian cycle and next by adding a self loop to each vertex. -The PRNG can thus be seen as a random walk of length in $\mathsf{P}$ +The PRNG can thus be seen as a random walk of length in $\mathcal{P}$ into this new $\mathsf{N}$-cube. -We furthermore have exhibited a bound on the number of iterations +We have exhibit an efficient method to compute such a balanced Hamiltonian +cycle. This method is an algebraic solution of an undeterministic +approach~\cite{ZanSup04} and has a low complexity. +Thanks to this solution, many chaotic functions can be generated. + + + +We furthermore have exhibited a upper bound on the number of iterations that is sufficient to obtain a uniform distribution of the output. +Such a upper bound is quadratic on the number of bits to output. +Experiments have however shown that such a bound is in +$\mathsf{N}.\log(\mathsf{N})$ in practice. + Finally, experiments through the NIST battery have shown that the statistical properties are almost established for -$\mathsf{N} = 4, 5, 6, 7, 8$. + $\mathsf{N} = 4, 5, 6, 7, 8$ and should be observed for any +positive integer $\mathsf{N}$. In future work, we intend to understand the link between statistical tests and the properties of chaos for @@ -29,6 +42,7 @@ 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