Efficient and portable multiple recursive generators of large order

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

[1]  Pierre L'Ecuyer,et al.  Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators , 1999, Oper. Res..

[2]  Pierre L'Ecuyer,et al.  An Implementation of the Lattice and Spectral Tests for Multiple Recursive Linear Random Number Generators , 1997, INFORMS J. Comput..

[3]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[4]  Pierre L'Ecuyer,et al.  A search for good multiple recursive random number generators , 1993, TOMC.

[5]  Pierre L'Ecuyer,et al.  Linear congruential generators of order K>1 , 1988, WSC '88.

[6]  Pierre L'Ecuyer,et al.  Bad Lattice Structures for Vectors of Nonsuccessive Values Produced by Some Linear Recurrences , 1997, INFORMS J. Comput..

[7]  Pierre L'Ecuyer,et al.  Efficient and portable combined random number generators , 1988, CACM.

[8]  Pei-Chi Wu,et al.  Multiplicative, congruential random-number generators with multiplier ± 2k1 ± 2k2 and modulus 2p - 1 , 1997, TOMS.

[9]  W. H. Payne,et al.  Coding the Lehmer pseudo-random number generator , 1969, CACM.

[10]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[11]  Dennis K. J. Lin,et al.  Random Number Generation for the New Century , 2000 .

[12]  Lih-Yuan Deng,et al.  Generalized Mersenne Prime Number and Its Application to Random Number Generation , 2004 .

[13]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[14]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[15]  Pierre L'Ecuyer,et al.  On the Deng-Lin random number generators and related methods , 2004, Stat. Comput..

[16]  Pierre L'Ecuyer,et al.  Beware of linear congruential generators with multipliers of the form a = ±2q ±2r , 1999, TOMS.

[17]  A. Grube,et al.  Mehrfach rekursiv‐erzeugte Pseudo‐Zufallszahlen , 1973 .

[18]  Lih-Yuan Deng,et al.  A system of high-dimensional, efficient, long-cycle and portable uniform random number generators , 2003, TOMC.

[19]  David Thomas,et al.  The Art in Computer Programming , 2001 .