Abstract
Random number generators based on linear recurrences modulo 2 are among the fastest long-period generators currently available. The uniformity and independence of the points they produce, by taking vectors of successive output values from all possible initial states, can be measured by theoretical figures of merit that can be computed quickly, and the generators having good values for these figures of merit are statistically reliable in general. Some of these generators can also provide disjoint streams and substreams efficiently. In this paper, we review the most interesting construction methods for these generators, examine their theoretical and empirical properties, describe the relevant computational tools and algorithms, and make comparisons.
Keywords
- Random Number Generator
- Characteristic Polynomial
- Combine Generator
- Minimal Polynomial
- Linear Feedback Shift Register
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
Brent, R. P. 2004. Note on Marsaglia’s xorshift random number generators. Journal of Statistical Software 11(5):1–4.
Brent, R. P., and P. Zimmermann. 2003. Random number generators with period divisible by a Mersenne prime. In Computational Science and its Applications—ICCSA 2003, eds. V. Kumar, M. L. Gavrilova, C. J. K. Tan, and P. L’Ecuyer, Volume 2667 of Lecture Notes in Computer Science, pp. 1–10. Berlin: Springer-Verlag.
Compagner, A. 1991. The hierarchy of correlations in random binary sequences. Journal of Statistical Physics 63:883–896.
Couture, R., and P. L’Ecuyer. 2000. Lattice computations for random numbers. Mathematics of Computation 69(230):757–765.
Erdmann, E. D. 1992. Empirical tests of binary keystreams. Master’s thesis, Department of Mathematics, Royal Holloway and Bedford New College, University of London.
Fishman, G. S. 1996. Monte Carlo: Concepts, Algorithms, and Applications. New York: Springer-Verlag.
Fushimi, M. 1983. Increasing the orders of equidistribution of the leading bits of the Tausworthe sequence. Information Processing Letters 16:189–192.
olub, G. H., and C. F. Van Loan. 1996. Matrix Computations, 3rd edition. Baltimore: John Hopkins University Press.
Haramoto, H., M. Matsumoto, T. Nishimura, F. Panneton, and P. L’Ecuyer. 2008. Efficient jump ahead for F 2-linear random number generators. INFORMS Journal on Computing 20(3):385–390.
Hellekalek, P., and G. Larcher. (eds.) 1998. Random and Quasi-Random Point Sets, Volume 138 of Lecture Notes in Statistics. New York: Springer-Verlag.
Knuth, D. E. 1998. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edition. Reading, MA: Addison-Wesley.
Law, A. M., and W. D. Kelton. 2000. Simulation Modeling and Analysis, 3rd edition. New York: McGraw-Hill.
L’Ecuyer, P. 1994. Uniform random number generation. Annals of Operations Research 53:77–120.
L’Ecuyer, P. 1996. Maximally equidistributed combined Tausworthe generators. Mathematics of Computation 65(213):203–213.
L’Ecuyer, P. 1999a. Good parameters and implementations for combined multiple recursive random number generators. Operations Research 47(1):159–164.
L’Ecuyer, P. 1999b. Tables of maximally equidistributed combined LFSR generators. Mathematics of Computation 68(225):261–269.
L’Ecuyer, P. 2004. Polynomial integration lattices. In Monte Carlo and Quasi-Monte Carlo Methods 2002, ed. H. Niederreiter, pp. 73–98. Berlin: Springer-Verlag.
L’Ecuyer, P. 2006. Uniform random number generation. In Simulation, eds. S. G. Henderson and B. L. Nelson, Handbooks in Operations Research and Management Science, pp. 55–81. Amsterdam, The Netherlands: Elsevier.
L’Ecuyer, P., and E. Buist. 2005. Simulation in Java with SSJ. In Proceedings of the 2005 Winter Simulation Conference, eds. M. E. Kuhl, N. M. Steiger, F. B. Armstrong, and J. A. Joines, pp. 611–620. Pistacaway, NJ: IEEE Press.
L’Ecuyer, P., and J. Granger-Piché. 2003. Combined generators with components from different families. Mathematics and Computers in Simulation 62:395–404.
L’Ecuyer, P., and C. Lemieux. 2002. Recent advances in randomized quasi-Monte Carlo methods. In Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, eds. M. Dror, P. L’Ecuyer, and F. Szidarovszky, pp. 419–474. Boston: Kluwer Academic.
L’Ecuyer, P., and F. Panneton. 2002. Construction of equidistributed generators based on linear recurrences modulo 2. In Monte Carlo and Quasi-Monte Carlo Methods 2000, eds. K.-T. Fang, F. J. Hickernell, and H. Niederreiter, pp. 318–330. Berlin: Springer-Verlag.
L’Ecuyer, P., and R. Simard. 2007, August. TestU01: A C library for empirical testing of random number generators. ACM Transactions on Mathematical Software 33(4):Article 22.
L’Ecuyer, P., R. Simard, E. J. Chen, and W. D. Kelton. 2002. An object-oriented random-number package with many long streams and substreams. Operations Research 50(6):1073–1075.
L’Ecuyer, P., and R. Touzin. 2000. Fast combined multiple recursive generators with multipliers of the form a = ± 2q ± 2r. In Proceedings of the 2000 Winter Simulation Conference, eds. J. A. Joines, R. R. Barton, K. Kang, and P. A. Fishwick, pp. 683–689. Pistacaway, NJ: IEEE Press.
Lemieux, C., and P. L’Ecuyer. 2002. Randomized polynomial lattice rules for multivariate integration and simulation. Extended version available online at http://www.iro.umontreal.ca/ lecuyer
Lemieux, C., and P. L’Ecuyer. 2003. Randomized polynomial lattice rules for multivariate integration and simulation. SIAM Journal on Scientific Computing 24(5):1768–1789.
Lenstra, A. K. 1985. Factoring multivariate polynomials over finite fields. Journal of Computer and System Sciences 30:235–248.
Lewis, T. G., and W. H. Payne. 1973. Generalized feedback shift register pseudorandom number algorithm. Journal of the ACM 20(3):456–468.
Lidl, R., and H. Niederreiter. 1986. Introduction to Finite Fields and Their Applications. Cambridge: Cambridge University Press.
Lindholm, J. H. 1968. An analysis of the pseudo-randomness properties of subsequences of long m-sequences. IEEE Transactions on Information Theory IT-14(4):569–576.
Marsaglia, G. 1985. A current view of random number generators. In Computer Science and Statistics, Sixteenth Symposium on the Interface, pp. 3–10. North-Holland, Amsterdam: Elsevier Science Publishers.
Marsaglia, G. 2003. Xorshift RNGs. Journal of Statistical Software 8(14):1–6.
Massey, J. L. 1969. Shift-register synthesis and BCH decoding. IEEE Transactions on Information Theory IT-15:122–127.
Matsumoto, M., and Y. Kurita. 1992. Twisted GFSR generators. ACM Transactions on Modeling and Computer Simulation 2(3):179–194.
Matsumoto, M., and Y. Kurita. 1994. Twisted GFSR generators II. ACM Transactions on Modeling and Computer Simulation 4(3):254–266.
Matsumoto, M., and Y. Kurita. 1996. Strong deviations from randomness in m-sequences based on trinomials. ACM Transactions on Modeling and Computer Simulation 6(2):99–106.
Matsumoto, M., and T. Nishimura. 1998. Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation 8(1):3–30.
Niederreiter, H. 1992. Random Number Generation and Quasi-Monte Carlo Methods, Volume 63 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: SIAM.
Nishimura, T. 2000. Tables of 64-bit Mersenne twisters. ACM Transactions on Modeling and Computer Simulation 10(4):348–357.
Panneton, F. 2004, August. Construction d’ensembles de points basée sur des récurrences linéaires dans un corps fini de caractéristique 2 pour la simulation Monte Carlo et l’intégration quasi-Monte Carlo. Ph.D. thesis, Département d’informatique et de recherche opérationnelle, Université de Montréal, Canada.
Panneton, F., and P. L’Ecuyer. 2004. Random number generators based on linear recurrences in \(F_{2^w}\). In Monte Carlo and Quasi-Monte Carlo Methods 2002, ed. H. Niederreiter, pp. 367–378. Berlin: Springer-Verlag.
Panneton, F., and P. L’Ecuyer. 2005. On the xorshift random number generators. ACM Transactions on Modeling and Computer Simulation 15(4): 346–361.
Panneton, F., and P. L’Ecuyer. 2007. Resolution-stationary random number generators. Mathematics and Computers in Simulation, to appear.
Panneton, F., P. L’Ecuyer, and M. Matsumoto. 2006. Improved long-period generators based on linear recurrences modulo 2. ACM Transactions on Mathematical Software 32(1):1–16.
Rieke, A., A.-R. Sadeghi, and W. Poguntke. 1998, August. On primitivity tests for polynomials. In Proceedings of the 1998 IEEE International Symposium on Information Theory. Cambridge, MA.
Strang, G. 1988. Linear Algebra and its Applications, 3rd edition. Philadelphia, PA: Saunders.
Tausworthe, R. C. 1965. Random numbers generated by linear recurrence modulo two. Mathematics of Computation 19:201–209.
Tezuka, S. 1995. Uniform Random Numbers: Theory and Practice. Norwell, MA: Kluwer Academic Publishers.
Tezuka, S., and P. L’Ecuyer. 1991. Efficient and portable combined Tausworthe random number generators. ACM Transactions on Modeling and Computer Simulation 1(2):99–112.
Tootill, J. P. R., W. D. Robinson, and D. J. Eagle. 1973. An asymptotically random Tausworthe sequence. Journal of the ACM 20:469–481.
Wang, D., and A. Compagner. 1993. On the use of reducible polynomials as random number generators. Mathematics of Computation 60:363–374.
Acknowledgements
This work has been supported by NSERC-Canada grant No. ODGP0110050 and a Canada Research Chair to the first author. This paper was written while the first author was a visiting scientist at the University of Salzburg, Austria, in 2005, and at IRISA, Rennes, in 2006. A short draft of it appeared in the Proceedings of the 2005 Winter Simulation Conference. Richard Simard helped doing the speed tests.
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L’Ecuyer, P., Panneton, F. (2009). F2-Linear Random Number Generators. In: Alexopoulos, C., Goldsman, D., Wilson, J. (eds) Advancing the Frontiers of Simulation. International Series in Operations Research & Management Science, vol 133. Springer, Boston, MA. https://doi.org/10.1007/b110059_9
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DOI: https://doi.org/10.1007/b110059_9
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