Periodic orbits of the ensemble of cat maps and pseudorandom number generation

We propose methods to construct high-quality pseudorandom number generators (RNG) 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, suppress these correlations dramatically and complicate deciphering. We analyze the mechanisms of the single-cat-map correlations analytically and show how to diminish them. We generalize PercivalVivaldi 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 found that the statistical properties of our best generators are not worse than that of other best modern generators.