Abstract
Statistical testing of pseudorandom number generators (PRNGs) is indispensable for their evaluation. A common difficulty among statistical tests is how we consider the resulting probability values (p-values). When we observe a small p-value such as 10−3, it is unclear whether it is due to a defect of the PRNG, or merely by chance. At the evaluation stage, we apply some hundred of different statistical tests to a PRNG. Even a good PRNG may produce some suspicious p-values in the results of a battery of tests. This may make the conclusions of the test battery unclear. This paper proposes an adaptive modification of statistical tests: once a suspicious p-value is observed, the adaptive statistical test procedure automatically increases the sample size, and tests the PRNG again. If the p-value is still suspicious, the procedure again increases the size, and re-tests. The procedure stops when the p-value falls either in an acceptable range, or in a clearly rejectable range. We implement such adaptive modifications of some statistical tests, in particular some of those in the Crush battery of TestU01. Experiments show that the evaluation of PRNGs becomes clearer and easier, and the sensitivity of the test is increased, at the cost of additional computation time.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
G. S. Fishman. Monte Carlo. Springer Series in Operations Research. Springer-Verlag, New York, 1996. Concepts, algorithms, and applications.
G. S. Fishman and L. R. Moore, III. An exhaustive analysis of multiplicative congruential random number generators with modulus 231−1. SIAM J. Sci. Statist. Comput., 7(3):1058, 1986.
F. James. RANLUX: a Fortran implementation of the high-quality pseudorandom number generator of Lüscher. Computer Physics Communications, 97:357–357(1), September 1996.
D. E. Knuth. The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1997.
P. L’Ecuyer. Testing random number generators. In WSC ’92: Proceedings of the 24th conference on Winter simulation, pages 305–313, New York, NY, USA, 1992. ACM.
P. L’Ecuyer. Tables of maximally equidistributed combined LFSR generators. Math. Comput., 68(225):261–269, 1999.
P. L’Ecuyer. Software for uniform random number generation: distinguishing the good and the bad. In WSC ’01: Proceedings of the 33nd conference on Winter simulation, pages 95–105, Washington, DC, USA, 2001. IEEE Computer Society.
P. L’Ecuyer, J. F. Cordeau, and R. Simard. Close-point spatial tests and their application to random number generators. Operations Research, 48(2):308–317, 2000.
P. L’Ecuyer and P. Hellekalek. Random number generators: selection criteria and testing. In Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 223–266. Springer, 1998.
P. L’Ecuyer and R. Simard. On the interaction of birthday spacings tests with certain families of random number generators. Mathematics and Computers in Simulation, 55:131–137, 2001.
P. L’Ecuyer and R. Simard. TestU01: a C library for empirical testing of random number generators. ACM Trans. Math. Software, 33(4):Art. 22, 40, 2007.
P. L’Ecuyer, R. Simard, and S. Wegenkittl. Sparse serial tests of uniformity for random number generators. SIAM Journal on Scientific Computing, 24(2):652–668, 2002.
P. C. Leopardi. Testing the tests: using pseudorandom number generators to improve empirical tests. Talk in MCQMC 2008, July 2008.
M. Lüscher. A portable high-quality random number generator for lattice field theory simulations. Comput. Phys. Comm., 79(1):100–110, 1994.
G. Marsaglia. A Current View of Random Number Generators. In Computer Science and Statistics, Sixteenth Symposium on the Interface, pages 3–10. Elsevier Science Publishers, 1985.
G. Marsaglia. DIEHARD: A battery of tests of randomness. 1996. See http://stat.fsu.edu/~geo/diehard.html.
G. Marsaglia. Xorshift RNGs. Journal of Statistical Software, 8(14):1–6, 2003.
G. Marsaglia, B. Narasimhan, and A. Zaman. A random number generator for PCs. Comput. Phys. Comm., 60(3):345–349, 1990.
M. Matsumoto and Y. Kurita. Twisted GFSR generators. ACM Trans. Model. Comput. Simul., 2(3):179–194, 1992.
M. Matsumoto and Y. Kurita. Twisted GFSR generators II. ACM Trans. Model. Comput. Simul., 4(3):254–266, 1994.
M. Matsumoto and T. Nishimura. Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul., 8(1):3–30, 1998.
M. Matsumoto and T. Nishimura. A nonempirical test on the weight of pseudorandom number generators. In Monte Carlo and quasi-Monte Carlo methods, 2000 (Hong Kong), pages 381–395. Springer, Berlin, 2002.
W. H. Press and S. A. Teukolsky. Numerical recipes in C (2nd ed.): the art of scientific computing. Cambridge University Press, New York, NY, USA, 1992.
A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo. A Statistical Test Suite for Random and Pseudorandom number Generators for Cryptographic Applications. NIST Special Publication 800-22, National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA, 2001. See http://csrc.nist.gov/rng/.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Haramoto, H. (2009). Automation of Statistical Tests on Randomness to Obtain Clearer Conclusion. In: L' Ecuyer, P., Owen, A. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04107-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-04107-5_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04106-8
Online ISBN: 978-3-642-04107-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
