X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/36e14b57e13803de510ec32c84df163f86dca6c3..3fc31015143d2bab7226a54390f3e1c5eba8f4d5:/conclusion.tex diff --git a/conclusion.tex b/conclusion.tex index e49ff17..6a24bc6 100644 --- a/conclusion.tex +++ b/conclusion.tex @@ -1,14 +1,14 @@ 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 +$p_1$ or $p_2$,\ldots, or $p_{\mathds{p}}$ 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 $\{p_1, \ldots, p_{\mathds{p}}\}$. -It can be deduced that in such a situation a PRNG, which iterates $G_f$ +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. +phenomena. We then have shown that a previously presented approach can be directly applied here to generate function $f$ with strongly connected @@ -21,18 +21,17 @@ into this new $\mathsf{N}$-cube. 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. -According to the author knowledge, this is the first time a full +According to the authors knowledge, this is the first time a full automatic method to provide chaotic PRNGs is given. Practically speaking, this approach preserves the security properties of the embedded PRNG, even if it remains quite cost expensive. -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. +We furthermore have exhibited an upper bound on the number of iterations +that is sufficient to obtain an uniform distribution of the output. +Such an 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 $\mathsf{N} = 4, 5, 6, 7, 8$ and should be observed for any @@ -44,7 +43,6 @@ 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