Pseudorandom Number Generation: Impossibility and Compromise

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Pseudorandom number generators are widely used in the area of simula- tion. Defective generators are still widely used in standard library programs, although better pseudorandom number generators such as the Mersenne Twister are freely avail- able. This manuscript gives a brief explanation on pseudorandom number generators for Monte Carlo simulation. The existing definitions of pseudorandomness are not sat- isfactorially 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 pseudorandom- ness, 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.