Efficient and portable multiple recursive generators of large order
Deng and Xu  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  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 1490301. © 2005 ACM.
ACM Transactions on Modeling and Computer Simulation
Deng, L. (2005). Efficient and portable multiple recursive generators of large order. ACM Transactions on Modeling and Computer Simulation, 15 (1), 1-13. https://doi.org/10.1145/1044322.1044323