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

Reviewer: William J. J. Rey

Multiplicative congruential generators are at the root of most numeric simulations, and the computers are always hungry for pseudorandom numbers. The 32-bit-based generators are still appropriate for most users, but there is a need to go to 64 bits. Of course, it is possible to combine generators, but that slows down the generation. According to the authors, this article presents the development of multiplicative congruential generators with prime modulus 2 p-1 and four forms of multipliers: ±2 k1±2 k2 . The multipliers for modulus 2 31-1 and 2 61-1 have been measured by spectral tests. The generators with these multipliers are portable and very fast. They have passed several empirical tests, including the frequency test, the run test, and the maximum-of-t test. In the future, more tests and more experience with these generators are needed (p. 256). The authors search for “good” multipliers. The method used is simple testing of whether a given multiplier is good. The author uses two kinds of guidelines to select the multipliers to be tested: the multiplier must satisfy given arithmetic rules, and it must allow a very efficient code for pseudorandom number generation. To the layperson, this method might appear rather crude; unfortunately, we do not yet have any safer method. This paper does not have much theoretical content. For those who need such multiplicative congruential generators, however, it reports coefficients they eagerly await. I continue to wonder about the usefulness of these generators when only short number sequences are needed. The asymptotic properties do not always tell the full story.