modif conclusion

diff --git a/conclusion.tex b/conclusion.tex
index 2705043fe2335fd223c705c26e86b2bc2409800a..0de1fc3040849c80f3e38688d96b00902ebfb2e3 100644 (file)
--- 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
 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 
 
 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. 
 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.
 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.
 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
 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


@@ -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. 
 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
 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