This work has assumed a Boolean map $f$ which is embedded into   
a discrete-time dynamical system $G_f$.
This one is supposed to be iterated a fixed number 
$p_1$ or $p_2$,\ldots, or $p_{\mathds{p}}$ of 
times before its output is considered. 
This work has first shown that iterations of
$G_f$ are chaotic if and only if its iteration graph $\Gamma_{\mathcal{P}}(f)$
is strongly connected where $\mathcal{P}$ is $\{p_1, \ldots, p_{\mathds{p}}\}$.
Any PRNG, which iterates $G_f$ as above 
satisfies in some cases the property of chaos.

We then have shown that a previously presented approach can be directly 
applied here to generate function $f$ with strongly connected 
$\Gamma_{\mathcal{P}}(f)$. 
The iterated map inside the generator is built by first removing from a 
$\mathsf{N}$-cube an Hamiltonian path and next 
adding  a self loop to each vertex. 
The PRNG can thus be seen as a random walks of length in $\mathsf{P}$
into  $\mathsf{N}$ this new cube.
We furthermore have exhibit a bound on the number of iterations 
that are sufficient to obtain a uniform distribution of the output.
Finally,  experiments through the  NIST battery have shown that
the statistical properties are almost established for
$\mathsf{N} = 4, 5, 6, 7, 8$.

In future work, we intend to understand the link between 
statistical tests and the properties of chaos for
the associated iterations.
By doing so, relations between desired statistically unbiased behaviors and
topological properties will be understood, leading to better choices
in iteration functions. 
Conditions allowing the reduction of the stopping-time will be
investigated too, while other modifications of the hypercube will
be regarded in order to enlarge the set of known chaotic
and random iterations.

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% ispell-dictionary: "american"
%%% mode: flyspell
%%% End:
