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.
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.
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}
-The next section presents how to build hamiltonian cycles in the
+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:
