X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/c069ead2bda7032833d2d7f3d0851906ec4d22f0..3fc31015143d2bab7226a54390f3e1c5eba8f4d5:/conclusion.tex?ds=inline

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 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
-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 
+phenomena.
 
 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 
-that is sufficient to obtain a uniform distribution of the output.
+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 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 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
-$\mathsf{N} = 4, 5, 6, 7, 8$.
+ $\mathsf{N} = 4, 5, 6, 7, 8$ 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