The lattice structure of pseudo-random number generators

The pairs, triples, etc. from most congruential pseudo-random number generators are known to lie on a lattice, and the ‘uniformity’ of these lattices is reflected in the quality of the output of the generator. Various characteristics of the lattices have been proposed as summaries of the quality of a generator, including the so-called lattice and spectral tests. This paper exploits the concept of polar lattices to show that these charac­terizations are essentially equivalent, and that they can be found to an approximation sufficient for assessing the quality of the generator without extensive searches. Checking generators is important, for many of those provided on small computers are inadequate for serious work.