X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/d69591c41135e899d27072006db6af016df62445..30a7ec2b1746fb3abfe8780a43625bb768842228:/generating.tex

diff --git a/generating.tex b/generating.tex
index 5b0d393..b4839db 100644
--- a/generating.tex
+++ b/generating.tex
@@ -3,16 +3,13 @@ It has been shown~\cite[Theorem 4]{bcgr11:ip} that
 if its iteration graph $\Gamma(f)$ is strongly connected, then 
 the output of $\chi_{\textit{14Secrypt}}$ follows 
 a law that tends to the uniform distribution 
-if and only if its Markov matrix is a doubly stochastic matrix.
-
-
+if and only if its Markov matrix is a doubly stochastic one.
 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.
 
-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. 
 
@@ -20,11 +17,11 @@ is removed, an edge $(x,x)$ is added.
 For instance, the iteration graph $\Gamma(f^*)$ 
 (given in Figure~\ref{fig:iteration:f*}) 
 is the $3$-cube in which the Hamiltonian cycle 
-$000,100,101,001,011,111,110,010,000$ 
+$000,100,101,001,011,111,$ $110,010,000$ 
 has been removed.
 \end{xpl}
 
-We first have proven the following result, which 
+We  have first proven the following result, which 
 states that the ${\mathsf{N}}$-cube without one
 Hamiltonian cycle 
 has the awaited property with regard to the connectivity.
@@ -32,10 +29,10 @@ has the awaited property with regard to the connectivity.
 \begin{thrm}
 The iteration graph $\Gamma(f)$ issued from
 the ${\mathsf{N}}$-cube where an Hamiltonian 
-cycle is removed is strongly connected.
+cycle is removed, is strongly connected.
 \end{thrm}
 
-Moreover, if all the transitions have the same probability ($\frac{1}{n}$),
+Moreover, when all the transitions have the same probability ($\frac{1}{n}$),
 we have proven the following results:
 \begin{thrm}
 The Markov Matrix $M$ resulting from the ${\mathsf{N}}$-cube in
@@ -46,21 +43,40 @@ 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.
-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 be solved 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 
-result.
+The answer is indeed positive. Furthermore, we have the following results which are stronger
+than previous ones. 
 \begin{thrm}
-There exist $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
+There exists $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
 \end{thrm}
 \begin{proof}
 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.
-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}
 
-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 as a way to build such codes, for instance 
+the Reflected Binary Code. In this one and 
+for a $\mathsf{N}$-length cycle, one of the bits is exactly switched 
+$2^{\mathsf{N}-1}$ times whereas the other bits are modified at most 
+$\left\lfloor \dfrac{2^{\mathsf{N-1}}}{\mathsf{N}-1} \right\rfloor$ times.
+It is clear that the function that is built from such a code would
+not provide a uniform output.  
+
+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: