Multiplicative, congruential random-number generators with multiplier ± 2 k 1 ± 2 k 2 and modulus 2 p - 1

Abstract

The demand for random numbers in scientific applications is increasing. However, the most widely used multiplicative, congruential random-number generators with modulus 2 31 - 1 have a cycle length of about 2.1 × 10 9 . Moreover, developing portable and efficient generators with a larger modulus such as 2 61 - 1 is more difficult than those with modulus 2 31 - 1. This article presents the development of multiplicative, congruential generators with modulus m = 2 p - 1 and four forms of multipliers: 2 k 1 - 2 k 2 , 2 k 1 + 2 k 2 , m - 2 k 1 + 2 k 2 , and m - 2 k 1 - 2 k 2 , k 1 > k 2. The multipliers for modulus 2 31 - 1 and 2 61 - 1 are measured by spectral tests, and the best ones are presented. The generators with these multipliers are portable and vary fast. They have also passed several empirical tests, including the frequency test, the run test, and the maximum-of-t test.