An exhaustive search for good 64-bit linear congruential random number generators with restricted multiplier

This paper explores that the different prime moduli can affect both the number of primitive root and the spectral test performance for 64-bit linear congruential generators (LCGs). Three forms of prime modulus and two types restriction on multiplier are considered in this paper. We perform computerized experiments that indicate significant differences exist among the number of primitive root of three forms of prime modulus. These differences can affect the performance of the spectral test. Two good 64-bit LCGs with significantly better spectral values and excellent empirical performance are presented. They are suitable for the requirements of todayʼs computer simulation studies.

► Different prime moduli can affect the number of primitive root.
► These differences lead to the larger spectral value of the LCGs with Sophie–Germain prime modulus.
► Two good 64-bit LCGs with sound lattice structure and excellent statistical performance are presented.