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[16dcc.git] / conclusion.tex

diff --git a/conclusion.tex b/conclusion.tex
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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 
 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
 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 
 
 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
 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
 
 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.
 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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