Improved long-period generators based on linear recurrences modulo 2

Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in this respect. In this article, we propose new generators of that form with better equidistribution and “bit-mixing” properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.

[1]  Makoto Matsumoto,et al.  Twisted GFSR generators II , 1994, TOMC.

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

[3]  Shu Tezuka,et al.  The k-distribution of generalized feedback shift register pseudorandom numbers , 1983, CACM.

[4]  Pierre L’Ecuyer,et al.  Random Number Generation , 2008, Encyclopedia of Algorithms.

[5]  R. Tausworthe.  Random Numbers Generated by Linear Recurrence Modulo Two , 1965 .

[6]  Pierre L'Ecuyer,et al.  Random Number Generators Based on Linear Recurrences in F 2 w , 2003 .

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

[8]  Pierre L'Ecuyer,et al.  Lattice computations for random numbers , 2000, Math. Comput..

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

[10]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[11]  S. Tezuka.  Uniform Random Numbers: Theory and Practice , 1995 .

[12]  Takuji Nishimura,et al.  Tables of 64-bit Mersenne twisters , 2000, TOMC.

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

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

[15]  J. P. R. Tootill,et al.  An Asymptotically Random Tausworthe Sequence , 1973, JACM.

[16]  Su. Centre-Ville,et al.  Construction of Equidistributed Generators based on linear recurrences modulo 2 , 2000 .

[17]  A. Compagner.  The hierarchy of correlations in random binary sequences , 1991 .

[18]  Aaldert Compagner,et al.  On the use of reducible polynomials as random number generators , 1993 .

[19]  Pierre L'Ecuyer,et al.  Efficient and portable combined Tausworthe random number generators , 1990, TOMC.

[20]  Pierre L'Ecuyer,et al.  Maximally equidistributed combined Tausworthe generators , 1996, Math. Comput..

[21]  Richard P. Brent,et al.  A fast algorithm for testing reducibility of trinomials mod~2 and some new primitive trinomials of degree 3021377 , 2003, Math. Comput..

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

[23]  Shu Tezuka,et al.  Uniform Random Numbers , 1995 .

[24]  Makoto Matsumoto,et al.  Strong deviations from randomness in m-sequences based on trinomials , 1996, TOMC.

[25]  James H. Lindholm.  An analysis of the pseudo-randomness properties of subsequences of long m -sequences , 1968, IEEE Trans. Inf. Theory.

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

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

[28]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[29]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1971, SYMSAC '71.

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

[31]  A. Rieke,et al.  On primitivity tests for polynomials , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[32]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

[33]  Francois Panneton.  Construction d'ensembles de points basee sur des recurrences lineaires dans un corps fini de caracteristique 2 pour la simulation Monte Carlo et l'integration quasi-Monte Carlo , 2004 .

[34]  Pierre L'Ecuyer,et al.  Construction of Equidistributed Generators Based on Linear Recurrences Modulo 2 , 2002 .

[35]  L'EcuyerPierre,et al.  Efficient and portable combined Tausworthe random number generators , 1991 .