Implementing 64-bit Maximally Equidistributed $\mathbb{F}_2$-Linear Generators with Mersenne Prime Period

TL;DR

This paper introduces 64-bit maximally equidistributed pseudorandom number generators with Mersenne prime periods, optimized for modern 64-bit architectures and matching the speed of existing Mersenne Twisters.

Contribution

It develops new 64-bit generators with optimal equidistribution properties and provides specific parameters for various large periods, improving upon existing generators.

Findings

Generators achieve maximal equidistribution in 64-bit space.

Speeds comparable to 64-bit Mersenne Twisters.

Provides parameter tables for periods up to 2^{44497}-1.

Abstract

CPUs and operating systems are moving from 32 to 64 bits, and hence it is important to have good pseudorandom number generators designed to fully exploit these word lengths. However, existing 64-bit very long period generators based on linear recurrences modulo 2 are not completely optimized in terms of the equidistribution properties. Here we develop 64-bit maximally equidistributed pseudorandom number generators that are optimal in this respect and have speeds equivalent to 64-bit Mersenne Twisters. We provide a table of specific parameters with period lengths from $2^{607}-1$ to $2^{44497}-1$. (An online appendix is available at http://www.ritsumei.ac.jp/~harase/melg-64-app.pdf)