Parallel Pseudorandom Number Generation Using Additive Lagged-Fibonacci Recursions

We study the suitability of the additive lagged-Fibonacci pseudorandom number generator for parallel computation. This generator has a relatively short period with respect to the size of its seed. However, the short period is more than made up for with the huge number of full-period cycles it contains. We call these different full-period cycles equivalence classes. We show how to enumerate the equivalence classes and how to compute seeds to select a given equivalence class. The use of these equivalence classes gives an explicit parallelization suitable for a fully reproducible asynchronous MIMD implementation. To explore such an implementation we introduce an exponential sum measure of quality for the additive lagged-Fibonacci generators used in serial or parallel. We then prove the first non-trivial results we are aware of on this measure of quality.