On the Link between Strongly Connected Iteration Graphs and Chaotic Boolean Discrete-Time Dynamical Systems

Abstract

Chaotic functions are characterized by sensitivity to initial conditions, transitivity, and regularity. Providing new functions with such properties is a real challenge. This work shows that one can associate with any Boolean network a continuous function, whose discrete-time iterations are chaotic if and only if the iteration graph of the Boolean network is strongly connected. Then, sufficient conditions for this strong connectivity are expressed on the interaction graph of this network, leading to a constructive method of chaotic function computation. The whole approach is evaluated in the chaos-based pseudo-random number generation context.

Efficient and cryptographically secure generation of chaotic pseudorandom numbers on GPU

“…The relation between ( f ) and G f is clear: there exists a path from x to x in ( f ) if and only if there exists a strategy s such that the parallel iteration of G f from the initial point (s, x) reaches the point x . We have then proven in [1] that, Devaney) …”

“…On the one hand, a post-treatment based on a chaotic dynamical system can be applied to a statistically deflective PRNG, to improve its statistical properties. Such an improvement can be found, for instance, in [1,4]. On the other hand, chaos can be added to a fast, statistically perfect PRNG and/or a cryptographically secure one, in case where chaos can be of interest, only if these last properties are not lost during the proposed post-treatment.…”

“…In some fields such as in numerical simulations, speed is a strong requirement that is usually attained using parallel architectures. In that case, a recurrent problem B Raphaël Couturier raphael.couturier@univ-fcomte.fr 1 FEMTO-ST Institute, UMR 6174 CNRS, University of Bourgogne Franche Comte, Belfort, France is that a deflation of the statistical qualities is often reported, when the parallelization of a good PRNG is realized. This is why ad hoc PRNGs for each possible architecture must be found to achieve both speed and randomness.…”

“…, we have established in[1] that,Theorem 2 Let f : B n → B n , ( f ) its iteration graph,M its adjacency matrix and M a n × n matrix defined by M i j = 1 nM i j if i = j and M ii = 1 − 1 n n j=1, j =iM i j otherwise. If ( f ) is strongly connected, then the output of the PRNG detailed in Algorithm 1 follows a law that tends to the uniform distribution if and only if M is a double stochastic matrix.…”

Dynamics on Binary Relations over Topological Spaces

“…We will analyze hypercyclicity, topological transitivity, and topologically mixing properties of binary relations. It is worth mentioning that the study of dynamics over finite graphs has been recently considered by Bahi et al by setting links between Devaney chaos and strong connectivity in order to provide an algorithm for the generation of strongly connected graphs and to construct Pseudo Random Number Generators (PRNGs) [6]. This approach has also allowed for the obtainment of PRNGs based on the construction of Hamiltonian cycles over an N-cube [7,8].…”