An experimental exploration of Marsaglia's xorshift Generators, Scrambled

Marsaglia proposed xorshift generators are a class of very fast, good-quality pseudorandom number generators. Subsequent analysis by Panneton and L'Ecuyer has lowered the expectations raised by Marsaglia's article, showing several weaknesses of such generators. Nonetheless, many of the weaknesses of xorshift generators fade away if their result is scrambled by a nonlinear operation (as originally suggested by Marsaglia). In this article we explore the space of possible generators obtained by multiplying the result of a xorshift generator by a suitable constant. We sample generators at 100 points of their state space and obtain detailed statistics that lead us to choices of parameters that improve on the current ones. We then explore for the first time the space of high-dimensional xorshift generators, following another suggestion in Marsaglia's article, finding choices of parameters providing periods of length 21024 - 1 and 24096 - 1. The resulting generators are of extremely high quality, faster than current similar alternatives, and generate long-period sequences passing strong statistical tests using only eight logical operations, one addition, and one multiplication by a constant.