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 $p_{\mathds{p}}$
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 It can be deduced that in such a situation a PRNG, which iterates $G_f$,
10 satisfies the property of chaos and can be used in simulating chaos
13 We then have shown that a previously presented approach can be directly
14 applied here to generate function $f$ with strongly connected
15 $\Gamma_{\mathcal{P}}(f)$.
16 The iterated map inside the generator is built by first removing from a
17 $\mathsf{N}$-cube a balanced Hamiltonian cycle and next
18 by adding a self loop to each vertex.
19 The PRNG can thus be seen as a random walk of length in $\mathcal{P}$
20 into this new $\mathsf{N}$-cube.
21 We have presented an efficient method to compute such a balanced Hamiltonian
22 cycle. This method is an algebraic solution of an undeterministic
23 approach~\cite{ZanSup04} and has a low complexity.
24 To the best of the authors knowledge, this is the first time a full
25 automatic method to provide chaotic PRNGs is given.
26 Practically speaking, this approach preserves the security properties of
27 the embedded PRNG, even if it remains quite cost expensive.
30 We furthermore have presented an upper bound on the number of iterations
31 that is sufficient to obtain an uniform distribution of the output.
32 Such an upper bound is quadratic on the number of bits to output.
33 Experiments have however shown that such a bound is in
34 $\mathsf{N}.\log(\mathsf{N})$ in practice.
35 Finally, experiments through the NIST battery have shown that
36 the statistical properties are almost established for
37 $\mathsf{N} = 4, 5, 6, 7, 8$ and should be observed for any
38 positive integer $\mathsf{N}$.
40 In future work, we intend to understand the link between
41 statistical tests and the properties of chaos for
42 the associated iterations.
43 By doing so, relations between desired statistically unbiased behaviors and
44 topological properties will be understood, leading to better choices
45 in iteration functions.
46 Conditions allowing the reduction of the stopping-time will be
47 investigated too, while other modifications of the hypercube will
48 be regarded in order to enlarge the set of known chaotic
49 and random iterations.
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