Spectrum and entropy of C-systems MIXMAX random number generator

Abstract

The uniformly hyperbolic Anosov C-systems defined on a torus have very strong instability of their trajectories, as strong as it can be in principle. These systems have exponential instability of all their trajectories and as such have mixing of all orders, nonzero Kolmogorov entropy and a countable set of everywhere dense periodic trajectories. In this paper we are studying the properties of their spectrum and of the entropy. For a two-parameter family of C-system operators A(N, s), parametrised by the integers N and s, we found the universal limiting form of the spectrum, the dependence of entropy on N and the period of its trajectories on a rational sublattice. One can deduce from this result that the entropy and the periods are sharply increasing with N . We present a new three-parameter family of C-operators A (N, s, m) and analyse the dependence of its spectrum and of the entropy on the parameter m. We are developing our earlier suggestion to use these tuneable Anosov C-systems for multipurpose Monte-Carlo simulations. The MIXMAX family of random number generators based on Anosov C-systems provide high quality statistical properties, thanks to their large entropy, have the best combination of speed, reasonable size of the state, tuneable parameters and availability for implementing the parallelisation.arXiv:1510.06274v3 [math.DS]

Review of High-Quality Random Number Generators

“…The simplest modification which increases the entropy is to use much larger values of the magic integer s in (3) up to the largest integers allowed by the MIXMAX N decimation 10 14 16 11 44 8 88 6 256 5 1000 4 computer arithmetic. This is called in [17] the two-parameter family A(N, s) with large s. A further modification was introduced using a new integer m to produce a new matrix A which is called the three-parameter family A(N, s, m) (see eq. 5 ),…”

“…The main properties of the new generator were published in [17] for selected values of (N, s, m) for which the period could be determined and which would be of most interest. Some of the results were nothing short of spectacular.…”

“…The obvious suspect is the large values of some elements of A which have caused the extended algorithm to lose the mixing properties of the continuous K-system. One upper limit on the allowed magnitude of elements of A is given by the developers of MIXMAX in [17]:…”

A comparative study and analysis of some pseudorandom number generator algorithms

“…The initial seed value u 0 may be a vector with a non‐zero component. An efficient implementation is presented in, with size of the matrix being 240, special entry in the matrix is 487 013 230 256 099 140, special multiplier is m = 2 51 + 1. We have performed NIST Test Suite on the MIXMAX PRNG and the results are presented in the Table .…”

The gonihedric paradigm extension of the Ising model

“…The phase structure of the spin systems can be studied by using Monte-Carlo simulations [59,60,61]. The random surfaces with area action in four dimensions are defined through ) is a function of temperature β and of the coupling constant k. At k = 0 the system (2.5), (2.6) reduces to (3.9), (4.18) and (4.22) with transfer matrix (4.23) and demonstrates, a strong first order phase transition.…”