Random Number Generators and Empirical Tests

  • Pierre L’Ecuyer
Part of the Lecture Notes in Statistics book series (LNS, volume 127)


We recall some requirements for “good” random number generators and argue that while the construction of generators and the choice of their parameters must be based on theory, a posteriori empirical testing is also important. We then give examples of tests failed by some popular generators and examples of generators passing these tests.


Random Number Generator Period Length Pseudorandom Number Output Sequence Uniform Random Variable 
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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  1. [BBS86]
    L. Blum, M. Blum, and M. Schub. A simple unpredictablepseudo-random number generator. SIAM Journal on Computing, 15(2):364–383, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [BFS87]
    P. Bratley, B. L. Fox, and L. E. Schrage. A Guide to Simu-lation. Springer-Verlag, New York, second edition, 1987.CrossRefGoogle Scholar
  3. [Com91]
    A. Compagner. The hierarchy of correlations in random bi-nary sequences. Journal of Statistical Physics, 63:883–896, 1991.CrossRefMathSciNetGoogle Scholar
  4. [Com95]
    A. Compagner. Operational conditions for random number generation. Physical Review E, 52(5-B):5634–5645, 1995.CrossRefGoogle Scholar
  5. [Dur73]
    J. Durbin. Distribution Theory for Tests Based on the Sam-ple Distribution Function, volume 9 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1973.CrossRefGoogle Scholar
  6. [DvdMST95]
    E. J. Dudewicz, E. C. van der Meulen, M. G. SriRam, and N. K. W. Teoh. Entropy-based random number evaluation. American Journal of Mathematical and Management Sciences, 15:115–153, 1995.CrossRefzbMATHGoogle Scholar
  7. [EG92]
    J. Eichenauer-Herrmann and H. Grothe. A new inversive congruential pseudorandom number generator with power of two modulus. ACM Transactions on Modeling and Computer Simulation, 2(1):1–11, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [EI94]
    J. Eichenauer-Herrmann and K. Ickstadt. Explicit inversive congruential pseudorandom numbers with power of two modulus. Mathematics of Computation, 62(206):787–797, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [Eic92]
    J. Eichenauer-Herrmann. Inversive congruential pseudoran-dom numbers: A tutorial. International Statistical Reviews, 60:167–176, 1992.CrossRefzbMATHGoogle Scholar
  10. [Eic96]
    J. Eichenauer-Herrmann. Modified explicit inversive congru-ential pseudorandom numbers with power-of-two modulus. Statistics and Computing, 6:31–36, 1996.CrossRefGoogle Scholar
  11. [FM86]
    G. S. Fishman and L. S. Moore III. An exhaustive analysis of multiplicative congruential random number generators with modulus 231 — 1. SIAM Journal on Scientific and Statistical Computing, 7(1):24–45, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [He195]
    P. Hellekalek. Inversive pseudorandom number generators:Concepts, results, and links. In C. Alexopoulos, K. Kang, W. R. Lilegdon, and D. Goldsman, editors, Proceedings of the 1995 Winter Simulation Conference, pages 255–262. IEEE Press, 1995.Google Scholar
  13. [Knu81]
    D. E. Knuth. The Art of Computer Programming, Vol-ume 2: Seminumerical Algorithms. Addison-Wesley, Reading, Mass., second edition, 1981.Google Scholar
  14. [Lag93]
    J. C. Lagarias. Pseudorandom numbers. Statistical Science,8(1):31–39, 1993.CrossRefMathSciNetGoogle Scholar
  15. [LCC96]
    P. L’Ecuyer, A. Compagner, and J.-F. Cordeau. Entropy-based tests for random number generators. Submitted. Also GERAD technical report number G-96–41, 1996.Google Scholar
  16. [LCS96]
    P. L’Ecuyer, J.-F. Cordeau, and R. Simard. Close-neighbor tests for random number generators. In preparation, 1996.Google Scholar
  17. [L’E88]
    P. L’Ecuyer. Efficient and portable combined random num-ber generators. Communications of the ACM, 31(6):742–749 and 774, 1988. See also the correspondence in the same journal, 32, 8 (1989) 1019–1024.Google Scholar
  18. [L’E92]
    P. L’Ecuyer. Testing random number generators. In Pro-ceedings of the 1992 Winter Simulation Conference, pages 305–313. IEEE Press, Dec 1992.Google Scholar
  19. [L’E94]
    P. L’Ecuyer. Uniform random number generation. Annals of Operations Research, 53:77–120, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [L’E96a]
    P. L’Ecuyer. Combined multiple recursive generators. Oper-ations Research, 44(5):816–822, 1996.CrossRefzbMATHGoogle Scholar
  21. [L’E96b]
    P. L’Ecuyer. TestU01: Un logiciel pour appliquer des tests statistiques à des générateurs de valeurs aléatoires. In preparation, 1996.Google Scholar
  22. [L’E97]
    P. L’Ecuyer. Random number generation. In Jerry Banks,editor, Handbook on Simulation. Wiley, 1997. To appear.Also GERAD technical report number G-96–38.Google Scholar
  23. [LK91]
    A. M. Law and W. D. Kelton. Simulation Modeling and Analysis. McGraw-Hill, New York, second edition, 1991.Google Scholar
  24. [LP89]
    P. L’Ecuyer and R. Proulx. About polynomial-time “unpre-dictable” generators. In Proceedings of the 1989 Winter Simulation Conference, pages 467–476. IEEE Press, Dec 1989.Google Scholar
  25. [LW97]
    H. Leeb and S. Wegenkitti. Inversive and linear congruentialpseudorandom number generators in selected empirical tests. ACM Transactions on Modeling and Computer Simulation, 1997. To appear.Google Scholar
  26. [Mar85]
    G. Marsaglia. A current view of random number generators.In Computer Science and Statistics, Sixteenth Symposium on the Interface, pages 3–10, North-Holland, Amsterdam, 1985. Elsevier Science Publishers.Google Scholar
  27. [Mar96]
    G. Marsaglia. Diehard: A battery of tests of randomness. 1996.Google Scholar
  28. [Nie92]
    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1992.CrossRefGoogle Scholar
  29. [Pla92]
    P. J. Plauger. The Standard C Library. Prentice Hall, En-glewood Cliffs, New Jersey, 1992.Google Scholar
  30. [Rip87]
    B. D. Ripley. Stochastic Simulation. Wiley, New York, 1987.CrossRefzbMATHGoogle Scholar
  31. [Rip90]
    B. D. Ripley. Thoughts on pseudorandom number genera-tors. Journal of Computational and Applied Mathematics, 31:153–163, 1990.CrossRefzbMATHGoogle Scholar
  32. [RS78]
    B. D. Ripley and B. W. Silverman. Quick tests for spatial interaction. Biometrika, 65(3):641–642, 1978.CrossRefGoogle Scholar
  33. [SB78]
    B. Silverman and T. Brown. Short distances, flat triangles and Poisson limits. Journal of Applied Probability, 15:815825, 1978.MathSciNetGoogle Scholar
  34. [Ste86a]
    M. S. Stephens. Tests based on EDF statistics. In R. B.D. Agostino and M. S. Stephens, editors, Goodness-of-Fit Techniques. Marcel Dekker, New York and Basel, 1986.Google Scholar
  35. [Ste86b]
    M. S. Stephens. Tests for the uniform distribution. In R. B. D. Agostino and M. S. Stephens, editors, Goodnessof-Fit Techniques, pages 331–366. Marcel Dekker, New York and Basel, 1986.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Pierre L’Ecuyer
    • 1
  1. 1.Département d’informatique et de recherche opérationnelleUniversité de MontréalMontréalCanada

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