Pseudorandom Number Generation: Impossibility and Compromise

Pseudorandom number generators are widely used in the area of simulation. Defective generators are still widely used in standard library programs, although better pseudorandom number generators such as the Mersenne Twister are freely available. This manuscript gives a brief explanation on pseudorandom number generators for Monte Carlo simulation. The existing definitions of pseudorandomness are not satisfactorially practical, since the generation of sequences satisfying the definitions is sometimes impossible, somtimes rather slow. As a compromise, to design a fast and reliable generator, some mathematical indices are used as measures of pseudorandomness, such as the period and the higher-dimensional equidistribution property. There is no rigorous justification for the use of these indices as measures of pseudorandomness, but experiences show their usefulness in choosing pseudorandom number generators.