On the Lattice Structure of a Special Class of Multiple Recursive Random Number Generators

We examine some properties of the points produced by certain classes of long-period linear multiple recursive random number generators proposed by L.-Y. Deng and his co-authors in several papers. These generators have their parameters selected in special ways to make the implementation faster. We show that as a result, the points produced by these generators have a poor lattice structure, and a poor initialization of the state can have long-lasting impact, because of the limited diffusion capacity of the recurrence.

[1]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

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

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

[4]  Lih-Yuan Deng,et al.  Scalable parallel multiple recursive generators of large order , 2009, Parallel Comput..

[5]  Makoto Matsumoto,et al.  Common defects in initialization of pseudorandom number generators , 2007, TOMC.

[6]  Pierre L'Ecuyer,et al.  An Object-Oriented Random-Number Package with Many Long Streams and Substreams , 2002, Oper. Res..

[7]  Pierre L'Ecuyer,et al.  Tables of maximally equidistributed combined LFSR generators , 1999, Math. Comput..

[8]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[9]  Pierre L'Ecuyer,et al.  TestU01: A C library for empirical testing of random number generators , 2006, TOMS.

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

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

[12]  Pierre L'Ecuyer,et al.  Combined Multiple Recursive Random Number Generators , 1995, Oper. Res..

[13]  Tony van Ravenstein.  The Three Gap Theorem (Steinhaus Conjecture) , 1988, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[14]  Pierre L'Ecuyer,et al.  Fast combined multiple recursive generators with multipliers of the form a = ±2q ±2r , 2000, WSC '00.

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

[16]  Pierre L'Ecuyer,et al.  Tables of linear congruential generators of different sizes and good lattice structure , 1999, Math. Comput..

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

[18]  P. L'Ecuyer,et al.  Fast combined multiple recursive generators with multipliers of the form a=/spl plusmn/2/sup q//spl plusmn/2/sup r/ , 2000, 2000 Winter Simulation Conference Proceedings (Cat. No.00CH37165).

[19]  Lih-Yuan Deng,et al.  Large-Order Multiple Recursive Generators with Modulus 231 - 1 , 2012, INFORMS J. Comput..

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

[21]  Lih-Yuan Deng,et al.  Efficient and portable multiple recursive generators of large order , 2005, TOMC.

[22]  Pierre L'Ecuyer,et al.  On the performance of birthday spacings tests with certain families of random number generators , 2001 .

[23]  Lih-Yuan Deng,et al.  Issues on Computer Search for Large Order Multiple Recursive Generators , 2008 .

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

[25]  P. A. W. Lewis,et al.  A Pseudo-Random Number Generator for the System/360 , 1969, IBM Syst. J..

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

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

[28]  Pierre L'Ecuyer,et al.  Uniform random number generation , 1994, Ann. Oper. Res..

[29]  Lih-Yuan Deng,et al.  Design and Implementation of Efficient and Portable Multiple Recursive Generators with Few Zero Coefficients , 2008 .

[30]  Pierre L'Ecuyer.  Uniform random number generators , 1998, WSC '98.