On the Distribution and Lattice Structure of Nonlinear Congruential Pseudorandom Numbers

Abstract

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers and a result on the s-dimensional lattice structure.1999 Academic Press

On the distribution of the power generator

“…We remark that for small e some nontrivial results can be derived from the estimates of the paper [24] which, although rather weak, apply to general polynomial generators.…”

“…In [24] the above bound has been used to obtain a nontrivial estimate for very general pseudorandom number generators. Nevertheless, for our purposes a direct application of the bound would give us rather weak results, the problem being that the dependence therein on the degree is not very good and the polynomials to which our argument naturally leads are in some cases of very high degree.…”

“…This approach is also used to study the distribution of several consecutive terms of the Blum-Blum-Shub generator. In some sense this approach is based on a combination of some ideas of [12,28] with a general approach to studying non-linear pseudorandom number generators invented in the series of papers [13,14,23,24,25,26].…”

Incomplete character sums and a special class of permutations

“…Consider the innermost sum on the right-hand side of (8). If applications to combinatorial design theory (see [2] for such applications).…”

On the Distribution of the Power Generator over a Residue Ring for Parts of the Period

“…Here we show that the original method of [20], and more recently also used in [8,9], combined with bounds for exponential sums with sparse polynomials from [12] allows us to study the distribution of the power generator of pseudorandom numbers over a residue ring. In particular, in [12] a distribution result for the sequence generated by (1) has been established for the sequence over the entire period.…”