Periodic orbits of the ensemble of Sinai-Arnold cat maps and pseudorandom number generation

Abstract

We propose methods for constructing high-quality pseudorandom number generators (RNGs) based on an ensemble of hyperbolic automorphisms of the unit two-dimensional torus (Sinai-Arnold map or cat map) while keeping a part of the information hidden. The single cat map provides the random properties expected from a good RNG and is hence an appropriate building block for an RNG, although unnecessary correlations are always present in practice. We show that introducing hidden variables and introducing rotation in the RNG output, accompanied with the proper initialization, dramatically suppress these correlations. We analyze the mechanisms of the single-cat-map correlations analytically and show how to diminish them. We generalize the Percival-Vivaldi theory in the case of the ensemble of maps, find the period of the proposed RNG analytically, and also analyze its properties. We present efficient practical realizations for the RNGs and check our predictions numerically. We also test our RNGs using the known stringent batteries of statistical tests and find that the statistical properties of our best generators are not worse than those of other best modern generators.

RNGSSELIB: Program library for random number generation, SSE2 realization

“…SSE2 technology allows to use 128-bit XMM-registers effectively and to accelerate computations. A similar technique was previously used for implementing some RNGs [9,6]. In the present work we demonstrate that this approach can essentially speed up most RNGs.…”

“…It is suggested in [6] to construct RNGs based on an ensemble of hyperbolic automorphisms of the unit twodimensional torus,…”

Applying dissipative dynamical systems to pseudorandom number generation: Equidistribution property and statistical independence of bits at distances up to logarithm of mesh size

“…hyperbolic automorphisms of the two-dimensional torus. In particular, it is proved in [4] that the probability of sequence 0000 of the first bits generated by a single cat map depends only on the trace k of a matrix M and for even k is equal to P = P 0 k 2 /(k 2 − 1), where P 0 = 1/16. If k is odd, then all sequences of length 4 are equiprobable.…”

“…These properties resemble certain properties of pseudorandomness and are considered in the literature as desirable properties for pseudorandomness. Several pseudorandom number generators based on chaotic maps have been proposed in the literature [3,4]. However, the behavior of dynamical systems on a discrete lattice is studied much less than in continuous space and a number of corresponding important questions still remain open.…”

“…[4,5], which is based on evolution of the ensemble of dynamical systems. Several generalizations are carried out.…”