Multiplicative, Congruential Random-Number Generators with Multiplier ± 2k1 ± 2k2 and Modulus 2p - 1

Abstract

The demand for random numbers in scientific applications is increasing. However, the most widely used multiplicative, congruential random-number generators with modulus 231 - 1 have a cycle length of about 2.1 × 109. Moreover, developing portable and efficient generators with a larger modulus such as 261 - 1 is more difficult than those with modulus 231 - 1. This article presents the development of multiplicative, congruential generators with modulus m = 2p - 1 and four forms of multipliers: 2k1 - 2k2, 2k1 + 2k2, m - 2k1 + 2k2, and m - 2k1 - 2k2, k1 > k2. The multipliers for modulus 231 - 1 and 261 - 1 are measured by spectral tests, and the best ones are presented. The generators with these multipliers are portable and very fast. They have also passed several empirical tests, including the frequency test, the run test, and the maximum-of-t test.