modif presentation

[16dcc.git] / conclusion.tex

diff --git a/conclusion.tex b/conclusion.tex
index e49ff176c5ddb133176714dc3343d5a3676db581..a656965cb88c88277d5cb99076cc4e0fe10f593e 100644 (file)
--- 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 
 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.

-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 
 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 
 
 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.
 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.
 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.
 
 
 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.
 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 
 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. 
 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