Multiplicative congruential random number generators with modulus 2^{𝛽}: an exhaustive analysis for 𝛽=32 and a partial analysis for 𝛽=48

Abstract

This paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 2 32 {2^{32}} and 2 48 {2^{48}} . Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k -tuples lie does not exceed the minimal achievable distance by more than 25 percent for k = 2 , … , 6 k = 2, \ldots ,6 . This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 132 multipliers out of the more than 536 million candidate multipliers that exist for modulus 2 32 {2^{32}} and to only 42 multipliers in a sample of about 67.1 million tested among the more than 351 × 10 11 351 \times {10^{11}} candidate multipliers for modulus 2 48 {2^{48}} . Section 1 reviews the basic properties of multiplicative congruential generators and § 2 \S 2 describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k -tuples and the discrepancy. For modulus 2 32 {2^{32}} , § 3 \S 3 presents the ten best multipliers and compares their performances with those of two multipliers that have been recommended in the literature. Comparisons using packing measures in the space of k -tuples and in the dual space are also made. For modulus 2 48 {2^{48}} , § 4 \S 4 also presents analogous results for the five best multipliers and for two multipliers suggested in the literature.