Efficient and portable multiple recursive generators of large order

Abstract

Deng and Xu [2003] proposed a system of multiple recursive generators of prime moduluspand orderk, where all nonzero coefficients of the recurrence are equal. This type of generator is efficient because only a single multiplication is required. It is common to choosep= 231−1 and some multipliers to further improve the speed of the generator. In this case, some fast implementations are available without using explicit division or multiplication. For such ap, Deng and Xu [2003] provided specific parameters, yielding the maximum period for recurrence of orderk, up to 120. One problem of extending it to a largerkis the difficulty of finding a complete factorization ofpk−1. In this article, we apply an efficient technique to findksuch that it is easy to factorpk−1, withp= 231−1. The largest one found isk= 1597. To find multiple recursive generators of large orderk, we introduce an efficient search algorithm with an early exit strategy in case of a failed search. Fork= 1597, we constructed several efficient and portable generators with the period length approximately 1014903.1.