pch

[16dcc.git] / generating.tex

diff --git a/generating.tex b/generating.tex
index 5b0d3931f3e51269806ef6df83a11a0731408304..d512a9803b59c27f0a95a6a6bff958b887a8f30d 100644 (file)
--- a/generating.tex
+++ b/generating.tex
@@ -7,12 +7,11 @@ if and only if its Markov matrix is a doubly stochastic matrix.
 
 
 In~\cite[Section 4]{DBLP:conf/secrypt/CouchotHGWB14},
 
 
 In~\cite[Section 4]{DBLP:conf/secrypt/CouchotHGWB14},

-we have presented an efficient
-approach which generates 
+we have presented a general scheme which generates 

 function with strongly connected iteration graph $\Gamma(f)$ and
 with doubly stochastic Markov probability matrix.
 
 function with strongly connected iteration graph $\Gamma(f)$ and
 with doubly stochastic Markov probability matrix.
 

-Basically, let consider the ${\mathsf{N}}$-cube. Let us next 
+Basically, let us consider the ${\mathsf{N}}$-cube. Let us next 

 remove one Hamiltonian cycle in this one. When an edge $(x,y)$ 
 is removed, an edge $(x,x)$ is added. 
 
 remove one Hamiltonian cycle in this one. When an edge $(x,y)$ 
 is removed, an edge $(x,x)$ is added. 
 


@@ -46,10 +45,10 @@ cycle is removed, is doubly stochastic.
 Let us consider now a ${\mathsf{N}}$-cube where an Hamiltonian 
 cycle is removed.
 Let $f$ be the corresponding function.
 Let us consider now a ${\mathsf{N}}$-cube where an Hamiltonian 
 cycle is removed.
 Let $f$ be the corresponding function.

-The question which remains to solve is
-can we always find $b$ such that $\Gamma_{\{b\}}(f)$ is strongly connected.
+The question which remains to solve is:
+\emph{can we always find $b$ such that $\Gamma_{\{b\}}(f)$ is strongly connected?}

 
 

-The answer is indeed positive. We furtheremore have the following strongest 
+The answer is indeed positive. We furthermore have the following strongest 

 result.
 \begin{thrm}
 There exist $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
 result.
 \begin{thrm}
 There exist $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.


@@ -59,8 +58,28 @@ There is an arc $(x,y)$ in the
 graph $\Gamma_{\{b\}}(f)$ if and only if $M^b_{xy}$ is positive
 where $M$ is the Markov matrix of $\Gamma(f)$.
 It has been shown in~\cite[Lemma 3]{bcgr11:ip}  that $M$ is regular.
 graph $\Gamma_{\{b\}}(f)$ if and only if $M^b_{xy}$ is positive
 where $M$ is the Markov matrix of $\Gamma(f)$.
 It has been shown in~\cite[Lemma 3]{bcgr11:ip}  that $M$ is regular.

-There exists thus $b$ such there is an arc between any $x$ and $y$.
+Thus, there exists $b$ such that there is an arc between any $x$ and $y$.

 \end{proof}
 
 \end{proof}
 

-Details on the construction of hamiltonian paths in the 
-$\mathsf{N}$-cube may be found in~\cite[Section 4]{DBLP:conf/secrypt/CouchotHGWB14}.
\ No newline at end of file
+This section ends with the idea of removing a Hamiltonian cycle in the 
+$\mathsf{N}$-cube. 
+In such a context, the Hamiltonian cycle is equivalent to a Gray code.
+Many approaches have been proposed a way to build such codes, for instance 
+the Reflected Binary Code. In this one, one of the bits is switched 
+exactly $2^{\mathsf{N}-}$ \ANNOT{formule incomplète : $2^{\mathsf{N}-1}$ ??} for a $\mathsf{N}$-length cycle. 
+
+%%%%%%%%%%%%%%%%%%%%%
+
+The function that is built 
+from the \ANNOT{Phrase non terminée}
+
+The next section presents how to build balanced Hamiltonian cycles in the 
+$\mathsf{N}$-cube with the objective to embed them into the 
+pseudorandom number generator.
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "main"
+%%% ispell-dictionary: "american"
+%%% mode: flyspell
+%%% End: