Combination of multiplicative congruential random-number generators with safe prime modulus

Abstract

Two or more multiplicative congruential random-number generators with prime modulus combined by means of a method proposed by Wichmann and Hill (1982) yield a random-number generator equivalent to a multiplicative congruential random-number generator with modulus equal to the product of the moduli of the component multiplicative congruential generators. The period of a random-number sequence obtained by the Wichmann-Hill method is equal to the least common multiple of the periods of the combined sequences. One of the two purposes of this paper is to present a necessary and sufficient set of efficiently verifiable conditions for the period to be equal to its maximum, which is the maximum of the least common multiple. Each of the conditions is always satisfied or is more easily verifiable when the modulus of each of the component generators is a safe prime. The other purpose of this paper is to derive an efficiently evaluatable formula for serial correlations of the maximum-period sequences by the Wichmann-Hill method. The authors recommend (i) to make the modulus of each of the component generators a safe prime, and (ii) to chose the multipliers of the components so as to (a) maximize the period and (b) make the serial correlations small in absolute value.