Secure pseudorandom bit generators and point sets with low star-discrepancy

Abstract

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasonable size with low discrepancy are badly needed. A seminal result from Heinrich, Novak, Wasilkowski and Woźniakowski shows the existence of a positive number C such that for every dimension d there exists an N-element point set in [0,1)d with star-discrepancy of at most Cd∕N. This is a pure existence result and explicit constructions of such point sets would be very desirable. The proofs are based on random samples of N-element point sets which are difficult to realize for practical applications.In this paper we propose to use secure pseudorandom bit generators for the generation of point sets with star-discrepancy of order O(d∕N). This proposal is supported theoretically and by means of numerical experiments.