Multiplicative, congruential random-number generators with multiplier ± 2^k1 ± 2^k2 and modulus 2^p - 1

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 = 2^p - 1 and four forms of multipliers: 2^k1 - 2^k2, 2^k1 + 2^k2, m - 2^k1 + 2^k2, and m - 2^k1 - 2^k2, 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 vary fast. They have also passed several empirical tests, including the frequency test, the run test, and the maximum-of-t test.