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
+$p_1$ 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 $\{p_1, \ldots, p_{\mathds{p}}\}$.
-Any PRNG, which iterates $G_f$ as above
-satisfies in some cases the property of chaos.
+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 an Hamiltonian path and next
+$\mathsf{N}$-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 $\mathsf{P}$
+The PRNG can thus be seen as a random walk of length in $\mathcal{P}$
into this new $\mathsf{N}$-cube.
-We furthermore have exhibited a bound on the number of iterations
-that is sufficient to obtain a uniform distribution of the output.
+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
-$\mathsf{N} = 4, 5, 6, 7, 8$.
+ $\mathsf{N} = 4, 5, 6, 7, 8$ 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
