1 This work has assumed a Boolean map $f$ which is embedded into
2 a discrete-time dynamical system $G_f$.
3 This one is supposed to be iterated a fixed number
4 $p_1$ or $p_2$,\ldots, or $p_{\mathds{p}}$ of
5 times before its output is considered.
6 This work has first shown that iterations of
7 $G_f$ are chaotic if and only if its iteration graph $\Gamma_{\mathcal{P}}(f)$
8 is strongly connected where $\mathcal{P}$ is $\{p_1, \ldots, p_{\mathds{p}}\}$.
9 Any PRNG, which iterates $G_f$ as above
10 satisfies in some cases the property of chaos.
12 We then have shown that a previously presented approach can be directly
13 applied here to generate function $f$ with strongly connected
14 $\Gamma_{\mathcal{P}}(f)$.
15 The iterated map inside the generator is built by first removing from a
16 $\mathsf{N}$-cube an Hamiltonian path and next
17 adding a self loop to each vertex.
18 The PRNG can thus be seen as a random walks of length in $\mathsf{P}$
19 into $\mathsf{N}$ this new cube.
20 We furthermore have exhibit a bound on the number of iterations
21 that are sufficient to obtain a uniform distribution of the output.
22 Finally, experiments through the NIST battery have shown that
23 the statistical properties are almost established for
24 $\mathsf{N} = 4, 5, 6, 7, 8$.
26 In future work, we intend to understand the link between
27 statistical tests and the properties of chaos for
28 the associated iterations.
29 By doing so, relations between desired statistically unbiased behaviors and
30 topological properties will be understood, leading to better choices
31 in iteration functions.
32 Conditions allowing the reduction of the stopping-time will be
33 investigated too, while other modifications of the hypercube will
34 be regarded in order to enlarge the set of known chaotic
35 and random iterations.
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