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Some Difficult-to-pass Tests of Randomness

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  • Marsaglia, George
  • Tsang, Wai Wan

Abstract

We describe three tests of randomness-- tests that many random number generators fail. In particular, all congruential generators-- even those based on a prime modulus-- fail at least one of the tests, as do many simple generators, such as shift register and lagged Fibonacci. On the other hand, generators that pass the three tests seem to pass all the tests in the Diehard Battery of Tests. Note that these tests concern the randomness of a generator's output as a sequence of independent, uniform 32-bit integers. For uses where the output is converted to uniform variates in [0,1), potential flaws of the output as integers will seldom cause problems after the conversion. Most generators seem to be adequate for producing a set of uniform reals in [0,1), but several important applications, notably in cryptography and number theory-- for example, establishing probable primes, complexity of factoring algorithms, random partitions of large integers-- may require satisfactory performance on the kinds of tests we describe here.

Suggested Citation

  • Marsaglia, George & Tsang, Wai Wan, 2002. "Some Difficult-to-pass Tests of Randomness," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 7(i03).
    • Handle: RePEc:jss:jstsof:v:007:i03
    DOI: http://hdl.handle.net/10.18637/jss.v007.i03
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    Cited by:

    1. Vladimir K. Kaishev & Dimitrina S. Dimitrova, 2009. "Dirichlet Bridge Sampling for the Variance Gamma Process: Pricing Path-Dependent Options," Management Science, INFORMS, vol. 55(3), pages 483-496, March.
    2. repec:jss:jstsof:21:i01 is not listed on IDEAS
    3. Castro, Julio Cesar Hernandez & Sierra, José María & Seznec, Andre & Izquierdo, Antonio & Ribagorda, Arturo, 2005. "The strict avalanche criterion randomness test," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(1), pages 1-7.
    4. Ilija Tanackov & Feta Sinani & Miomir Stanković & Vuk Bogdanović & Željko Stević & Mladen Vidić & Jelena Mihaljev-Martinov, 2019. "Natural Test for Random Numbers Generator Based on Exponential Distribution," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    5. Doyle, John R. & Chen, Catherine H., 2013. "Patterns in stock market movements tested as random number generators," European Journal of Operational Research, Elsevier, vol. 227(1), pages 122-132.
    6. Dlugosz, Stephan & Müller-Funk, Ulrich, 2012. "Ziffernanalyse zur Betrugserkennung in Finanzverwaltungen: Prüfung von Kassenbelegen," Arbeitsberichte des Instituts für Wirtschaftsinformatik 133, University of Münster, Department of Information Systems.
    7. Ben Moews, 2023. "On random number generators and practical market efficiency," Papers 2305.17419, arXiv.org, revised Jul 2023.
    8. Almaraz Luengo, Elena & Leiva Cerna, Marcos Brian & García Villalba, Luis Javier & Hernandez-Castro, Julio, 2022. "A new approach to analyze the independence of statistical tests of randomness," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    9. Altman, Micah & Gill, Jeff & McDonald, Michael P., 2007. "accuracy: Tools for Accurate and Reliable Statistical Computing," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i01).
    10. Madiha Khalid & Umar Mujahid & Muhammad Najam-ul-Islam, 2018. "Cryptanalysis of ultralightweight mutual authentication protocol for radio frequency identification enabled Internet of Things networks," International Journal of Distributed Sensor Networks, , vol. 14(8), pages 15501477187, August.

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