-Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$
+Let us finally present the pseudorandom number generator $\chi_{\textit{15Rairo}}$,
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 integer $b$ that corresponds to 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.},
+a random choice of which edge to follow, and traverses this edge
+provided it is allowed to do so, \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}.
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,
+Since the $\chi_{\textit{15Rairo}}$ algorithm
+only adds probability constraints on existing edges,
it preserves this property.
For each number $\mathsf{N}=4,5,6,7,8$ of bits, we have generated
-the functions according the method
-given in Sect.~\ref{sec:SCCfunc} .
+the functions according to the method
+given in Sect.~\ref{sec:SCCfunc}.
For each $\mathsf{N}$, we have then restricted this evaluation to the function
whose Markov Matrix (issued from Eq.~(\ref{eq:Markov:rairo}))
has the smallest practical mixing time.
the second list (namely~14).
In this table the column
-which is labeled with $b$ (respectively by $E[\tau]$)
+that is labeled with $b$ (respectively by $E[\tau]$)
gives the practical mixing time
-where the deviation to the standard distribution is less than $10^{-6}$
-(resp. the theoretical upper bound ofstopping time as described in
+where the deviation to the standard distribution is lesser than $10^{-6}$
+(resp. the theoretical upper bound of stopping time as described in
Sect.~\ref{sec:hypercube}).
and the generator is unsuitable. Table~\ref{The passing rate} shows $\mathbb{P}_T$ of sequences based on discrete
chaotic iterations using different schemes. If there are at least two statistical values in a test, this test is
marked with an asterisk and the average value is computed to characterize the statistics.
-We can see in Table \ref{The passing rate} that all the rates are greater than 97/100, \textit{i. e.}, all the generators pass the NIST test.
+We can see in Table \ref{The passing rate} that all the rates are greater than 97/100, \textit{i.e.}, all the generators
+achieve to pass the NIST battery of tests.
+
\begin{table}
