Mathematics and Computers in Simulation 55 (2001) 281–288 Gambling tests for pseudorandom number generators Stefan Wegenkittl ∗ Institut für Mathematik, Universität Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria Abstract This paper extends the idea of serial tests by employing a carefully selected dimension reduction which is equivalent to playing a gambling strategy in a fair coin flipping game. We apply the generalized φ-divergence for testing the hypothesis that the simulated coin is fair and memoryless. An application to twisted GFSR generators shows the ability of our test to detect deviations from equidistribution in high dimensions. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Gambling test; Pseudorandom number generator; Generalized phi-divergence; Twisted GFSR generator 1. Introduction Numerous tests have been suggested for the empirical quality assessment of pseudorandom number generators (PRNGs), see [5] for an introduction. The standard battery of tests including serial (relative frequency based) tests for overlapping and non-overlapping tuples, and run (permutation based) tests, has recently been extended towards random-walk simulation [3,13–17]. We somewhat follow this direction and consider a gambling policy in a simple fair coin flipping game. The formulation in terms of gambling being transparent and evident, our test also belongs, technically speaking, to the class of overlapping serial tests with high dimension and employs the technique of dimension reduction to bypass the explosion of memory (or time) complexity which would normally impede the execution of the test. This paper is organized as follows. In Section 2 we recall serial tests for PRNGs. We introduce the gambling test in Section 3 and consider the generalized φ ϕ -divergence in Section 4. Section 5 contains an empirical study which shows the ability of the Gambling test to reveal correlations in twisted GFSR generators and sheds some light on the role of ϕ in the generalized φ ϕ -divergence. Conclusions are drawn in Section 6 and Appendix A contains technical details. ∗ Research supported by FWR-Project P13480, FSP Number-Theoretic Algorithms and their applications. URL: www:http://random.mat.sbg.ac.at. 0378-4754/01/$20.00 © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0378-4754(00)00271-8