A Method of Systematic Search for Optimal Multipliers in Congruential Random Number Generators

This paper presents a method of systematic search for optimal multipliers for congruential random number generators. The word-size of computers is a limiting factor for development of random numbers. The generators for computers up to 32 bit word-size are already investigated in detail by several authors. Some partial works are also carried out for moduli of 2⁴⁸ and higher sizes. Rapid advances in computer technology introduced recently 64 bit architecture in computers. There are considerable efforts to provide appropriate parameters for 64 and 128 bit moduli. Although combined generators are equivalent to huge modulus linear congruential generators, for computational efficiency, it is still advisable to choose the maximum moduli for the component generators. Due to enormous computational price of present algorithms, there is a great need for guidelines and rules for systematic search techniques. Here we propose a search method which provides ‘fertile’ areas of multipliers of perfect quality for spectral test in two dimensions. The method may be generalized to higher dimensions. Since figures of merit are extremely variable in dimensions higher than two, it is possible to find similar intervals if the modulus is very large. This revised version was published online in July 2006 with corrections to the Cover Date.