Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$
-which is based on random walks in $\Gamma(f)$.
-More precisely, let be given a Boolean map $f:\Bool^n \rightarrow \Bool^n$,
+which is based on random walks in $\Gamma_{\{b\}}(f)$.
+More precisely, let be given a Boolean map $f:\Bool^{\mathsf{N}} \rightarrow
+\Bool^\mathsf{N}$,
a PRNG \textit{Random},
an integer $b$ that corresponds an iteration number (\textit{i.e.}, the length of the walk), and
an initial configuration $x^0$.
\end{algorithm}
-This PRNG is a particularized version of Algorithm given in~\cite{DBLP:conf/secrypt/CouchotHGWB14}.
+This PRNG is slightly different from $\chi_{\textit{14Secrypt}}$
+recalled in Algorithm~\ref{CI Algorithm}.
As this latter, the length of the random
walk of our algorithm is always constant (and is equal to $b$).
However, in the current version, we add the constraint that
the probability to execute the function $F_f$ is equal to 0.5 since
the output of $\textit{Random(1)}$ is uniform in $\{0,1\}$.
+This constraint is added to match the theoretical framework of
+Sect.~\ref{sec:hypercube}.
+
+
+
+Notice that the chaos property of $G_f$ given in Sect.\ref{sec:proofOfChaos}
+only requires that the graph $\Gamma_{\{b\}}(f)$ is strongly connected.
+Since the $\chi_{\textit{15Rairo}}$ algorithme
+only adds propbability constraints on existing edges,
+it preserves this property.
+
-Let $f: \Bool^{n} \rightarrow \Bool^{n}$.
-It has been shown~\cite[Th. 4, p. 135]{bcgr11:ip} that
-if its iteration graph 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.
-
-Let us now present a method to
-generate functions
-with Doubly Stochastic matrix and Strongly Connected iteration graph,
-denoted as DSSC matrix.
