Resolution-stationary random number generators

Besides speed and period length, the quality of uniform random number generators is usually assessed by measuring the uniformity of their point sets, formed by taking vectors of successive output values over their entire period length. For F2-linear generators, the commonly adopted measures of uniformity are based on the equidistribution of the most significant bits of the output. In this paper, we point out weaknesses of these measures and introduce generalizations that also give importance to the low-order (less significant) bits. These measures look at the equidistribution obtained when we permute the bits of each output value in a certain way. In a parameter search for good generators, a quality criterion based on these new measures of equidistribution helps avoiding generators that fail statistical tests targeting their low-order bits. We also introduce the notion of resolution-stationary generators, whose point sets are invariant under a multiplication by certain powers of 2, modulo 1. For such generators, less significant bits have the same equidistribution properties as the most significant ones. Tausworthe generators have this property. We finally show how an arbitrary F2-linear generator can be made resolution-stationary by adding an appropriate linear transformation to the output. This provides new efficient ways of implementing high-quality and long-period Tausworthe generators.