X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/d69591c41135e899d27072006db6af016df62445..151dc81946eee7c7fb5fbc327ada099f2af6a2c3:/conclusion.tex?ds=sidebyside

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+++ b/conclusion.tex
@@ -1,27 +1,41 @@
 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 
-adding  a self loop to each vertex. 
-The PRNG can thus be seen as a random walks of length in $\mathsf{P}$
-into  $\mathsf{N}$ this new cube.
-We furthermore have exhibit a bound on the number of iterations 
-that are sufficient to obtain a uniform distribution of the output.
+$\mathsf{N}$-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 $\mathcal{P}$
+into  this new $\mathsf{N}$-cube.
+We have presented 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.
+To the best of 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 presented 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
@@ -33,3 +47,10 @@ 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
 and random iterations.
+
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