On the Distribution of Counter-Dependent Nonlinear Congruential Pseudorandom Number Generators in Residue Rings

Abstract

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Shparlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudorandom numbers generated by counter-dependent generators modulo a composite M .

Exponential sums for nonlinear recurring sequences

“…The improved method was already used in [5] as well. More precisely, the papers [3,4] deal with nonlinear recurrences y n+1 ≡ f (y n ) mod M with composite moduli M. The papers [6,7] contain extensions of (2) to nonlinear recurrences of higher orders y n+1 = f (y n , .…”

“…, y n−m+1 ) for some m 1 (see also the recent paper [1]). In [5] sequences y n+1 = f (y n , n) are considered. Finally, in [13] we proved analogs of (2) for multiplicative character sums over finite fields for nonlinear recurring sequences and their applications to the distribution of powers and primitive roots in finite fields.…”