Modified explicit inversive congruential pseudorandom numbers with power of 2 modulus

A modified version of the explicit inversive congruential method with power of 2 modulus for generating uniform pseudorandom numbers is introduced. The statistical independence behaviour of the generated sequences is studied based on the distribution of all pairs of successive pseudorandom numbers over the entire period. Lower and upper bounds for the discrepancy of the corresponding two-dimensional point sets are established. These results certainly play only a minor part in studying the statistical independence behaviour of the generated sequences, but they show that modified explicit inversive congruential pseudorandom numbers have some attractive properties at least regarding their two-dimensional discrepancy. The method of proof relies heavily on a thorough analysis of certain exponential sums.