X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/36e14b57e13803de510ec32c84df163f86dca6c3..b67a97afb3770294e649ce1a1ed53af685be00b9:/conclusion.tex

diff --git a/conclusion.tex b/conclusion.tex
index e49ff17..a656965 100644
--- a/conclusion.tex
+++ b/conclusion.tex
@@ -1,14 +1,14 @@
 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
-It can be deduced that in such a situation a PRNG, which iterates $G_f$ 
+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.
+phenomena.
 
 We then have shown that a previously presented approach can be directly 
 applied here to generate function $f$ with strongly connected 
@@ -18,21 +18,20 @@ $\mathsf{N}$-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 exhibit an efficient method to compute such a balanced Hamiltonian 
+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.
-According to the author knowledge, this is the first time a full 
+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 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.
+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
  $\mathsf{N} = 4, 5, 6, 7, 8$ and should be observed for any 
@@ -44,7 +43,6 @@ 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