Combined generators with components from different families

Most random number generators used in practice are based on linear recurrences, with linear output transformations. This gives long periods, fast implementations, and structures that are easy to analyze. But the points produced by these generators have very regular structures. Nonlinear generators can have less regular structures, but they are generally slower and much harder to analyze when their period is long. In this paper, combined generators with one large linear component, and a second component of a different type (nonlinear or linear), are proposed and studied. The structure of vectors of successive and non-successive output values produced by the combined generators is analyzed. Under mild conditions, these vector sets are proved to have at least as much uniformity than the corresponding sets for the linear component alone. In empirical statistical tests, these combined generators perform better than simple linear generator of comparable period lengths, because of their less regular structure. Efficient implementation methods are suggested.