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 $p_{\mathds{p}}$ 
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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
It can be deduced that in such a situation a PRNG, which iterates $G_f$,
satisfies the property of chaos and can be used in simulating chaos 
phenomena.

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 a balanced  Hamiltonian cycle and next 
by adding  a self loop to each vertex. 
The PRNG can thus be seen as a random walk of length in $\mathcal{P}$
into  this new $\mathsf{N}$-cube.
We have presented an efficient method to compute such a balanced Hamiltonian 
cycle. This method is an algebraic solution of an undeterministic 
approach~\cite{ZanSup04} and has a low complexity.
To the best of the authors knowledge, this is the first time a full 
automatic method to provide chaotic PRNGs is given.  
Practically speaking, this approach preserves the security properties of 
the embedded PRNG, even if it remains quite cost expensive.


We furthermore have presented an upper bound on the number of iterations 
that is sufficient to obtain an uniform distribution of the output.
Such an upper bound is quadratic on the number of bits to output.
Experiments have however shown that such a bound is in 
$\mathsf{N}.\log(\mathsf{N})$ in practice.
Finally,  experiments through the  NIST battery have shown that
the statistical properties are almost established for
 $\mathsf{N} = 4, 5, 6, 7, 8$ and should be observed for any 
positive integer $\mathsf{N}$.

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.

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