Efficient and portable multiple recursive generators of large order

Abstract

Deng and Xu [2003] proposed a system of multiple recursive generators of prime modulus p and order k , 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 choose p = 2 31 −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 a p , Deng and Xu [2003] provided specific parameters, yielding the maximum period for recurrence of order k , up to 120. One problem of extending it to a larger k is the difficulty of finding a complete factorization of p k −1. In this article, we apply an efficient technique to find k such that it is easy to factor p k −1, with p = 2 31 −1. The largest one found is k = 1597. To find multiple recursive generators of large order k , we introduce an efficient search algorithm with an early exit strategy in case of a failed search. For k = 1597, we constructed several efficient and portable generators with the period length approximately 10 14903.1 .