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 $f:\Bool^n \rightarrow \Bool^n$,
a PRNG \textit{Random},
an integer $b$ that corresponds an iteration number (\textit{i.e.}, the length of the walk), and 
an initial configuration $x^0$. 
Starting from $x^0$, the algorithm repeats $b$ times 
a random choice of which edge to follow and traverses this edge 
provided it is allowed to traverse it, \textit{i.e.}, 
when $\textit{Random}(1)$ is not null. 
The final configuration is thus outputted.
This PRNG is formalized in Algorithm~\ref{CI Algorithm}.



\begin{algorithm}[ht]
%\begin{scriptsize}
\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)}
\KwOut{a configuration $x$ ($n$ bits)}
$x\leftarrow x^0$\;
\For{$i=0,\dots,b-1$}
{
\If{$\textit{Random}(1) \neq 0$}{
$s\leftarrow{\textit{Random}(n)}$\;
$x\leftarrow{F_f(s,x)}$\;
}
}
return $x$\;
%\end{scriptsize}
\caption{Pseudo Code of the $\chi_{\textit{15Rairo}}$ PRNG}
\label{CI Algorithm}
\end{algorithm}


This PRNG is a particularized version of Algorithm given in~\cite{DBLP:conf/secrypt/CouchotHGWB14}.
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 $\{0,1\}$.  

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.   
