A Unified Approach to the Analysis of Compound Pseudorandom Numbers

The present paper deals with a general compound method for generating uniform pseudorandom numbers. Equidistribution and statistical independence properties of the generated sequences are studied based on the discrepancy of certain point sets. A unified approach to the analysis of the full period and of (relatively large) parts of the period is worked out, which rests on bounds for certain exponential sums over finite fields. This calculus is applied to the compound nonlinear congruential method and to the compound explicit inversive congruential method, which have been introduced recently. Known upper bounds for the discrepancy over the full period are improved and new upper bounds for the discrepancy over parts of the period are established.