Spectral Test of the MIXMAX Random Number Generators

TL;DR

This paper analyzes the spectral properties of MIXMAX pseudo-random number generators in higher dimensions, studying how internal parameters affect their distribution and proposing optimized parameter ranges for improved randomness quality.

Contribution

The study investigates the dependence of MIXMAX generators' spectral properties on internal parameters, identifying optimal ranges for parameter m to enhance generator performance.

Findings

Optimal m range between 2^{24} and 2^{36} for best spectral properties.

Provided alternative parameters for N=8 and N=240 generators within the optimized m range.

Confirmed that spectral properties improve when m is within the identified optimal range.

Abstract

An important statistical test on the pseudo-random number generators is called the spectral test. The test is aimed at answering the question of distribution of the generated pseudo-random vectors in dimensions $d$ that are larger than the genuine dimension of a generator $N$. In particular, the default MIXMAX generators have various dimensions: $N=8,17,240$ and higher. Therefore the spectral test is important to perform in dimensions $d > 8$ for $N=8$ generator, $d> 17$ for $N=17$ and $d> 240$ for $N=240$ generator. These tests have been performed by L'Ecuyer and collaborators. When $d > N$ the vectors of the generated numbers fall into the parallel hyperplanes and the distances between them can be larger than the genuine "resolution" of the MIXMAX generators, which is $ l=2^{-61}$. The aim of this article is to further study the spectral properties of the MIXMAX generators, to investigate the dependence of the spectral properties of the MIXMAX generators as a function of their internal parameters and in particular their dependence on the parameter $m$. We found that the best spectral properties are realized when $m$ is between $2^{24}$ and $2^{36}$, a range which is inclusive of the value of the $N=17$ generator. We also provide the alternative parameters for the generators, $N=8$ and $N=240$ with $m$ in this optimised range.