Generalized Lehmer-Tausworthe random number generators

We study a general class of random number generators which includes Lehmer's congruential generator and the Tausworthe shift-register generator as special cases. The generators in this class use a general linear recurrence relation defined by a primitive polynomial over a large finite field. This generator, like the Tausworthe generator, has the property of the k-space equi-distribution. We give some theoretical and heuristic justification for its asymptotic uniformity as well as asymptotic independence from a statistical theory viewpoint. In this paper, we also propose an efficient method of finding primitive polynomials in a large finite field. Several generators with extremely long cycles are presented.

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