From: couchot <couchot@localhost.localdomain>
Date: Thu, 30 Jun 2016 12:46:24 +0000 (+0200)
Subject: modif conclusion
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/commitdiff_plain/789817c68132962fe415afd50e1105d04fabea47?hp=-c

modif conclusion
---

789817c68132962fe415afd50e1105d04fabea47
diff --git a/conclusion.tex b/conclusion.tex
index 2705043..0de1fc3 100644
--- 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
-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 
-$\mathsf{N}$-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 
+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.
+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
-$\mathsf{N} = 4, 5, 6, 7, 8$.
+ $\mathsf{N} = 4, 5, 6, 7, 8$ 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
@@ -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. 
+
 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