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The star-discrepancy is a quantitative measure for the irregularity ofdistribution of a point set in the unit cube that is intimately linked to theintegration error of quasi-Monte Carlo algorithms. These popular integrationrules are nowadays also applied to very high-dimensional integration problems.Hence multi-dimensional point sets of reasonable size with low discrepancy arebadly needed. A seminal result from Heinrich, Novak, Wasilkowski andWo\'{z}niakowski shows the existence of a positive number $C$ such that forevery dimension $d$ there exists an $N$-element point set in $[0,1)^d$ withstar-discrepancy of at most $C\sqrt{d/N}$. This is a pure existence result andexplicit constructions of such point sets would be very desirable. The proofsare based on random samples of $N$-element point sets which are difficult torealize for practical applications.  In this paper we propose to use secure pseudorandom bit generators for thegeneration of point sets with star-discrepancy of order $O(\sqrt{d/N})$. Thisproposal is supported theoretically and by means of numerical experiments.

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