Exponential sums for nonlinear recurring sequences

Abstract

We prove a new bound on exponential sums for nonlinear recurring sequences. This result improves on an earlier bound of Niederreiter and Shparlinski. An application to the distribution and statistical independence of nonlinear congruential pseudorandom numbers is given.

On the average distribution of pseudorandom numbers generated by nonlinear permutations

“…However, here we propose a different argument, which allows us to consider the case of nonlinear permutations, and in fact we also obtain an improvement of the result of [7] as well. These estimates for almost all initial values are much stronger and hold in a much wider range than those obtained for every initial value; see [6] and [10] for the case of inversive and polynomial generators, respectively. We note that besides being of theoretical interest, such results that apply to almost all initial values are probably most relevant for practical applications of pseudorandom generators, in both cryptography and numerical analysis.…”

Open Problems on Exponential and Character Sums

“…Comments: The bound of [100] develops some ideas suggested in [130] and also improves the previous result of [96]; however it is still nontrivial only if T is very close to its largest possible value p. Even constructing some special (but general enough) families of polynomials for which such improvement is possible is of interest. In the multidimensional case (that is, for iterations of multivariate polynomials) such families are constructed in [102], see also [103].…”

“…In fact, without loss of generality, one can assume that it is purely periodic, that is, N 0 = 0. Improve the bound of H. Niederreiter and A. Winterhof [100] on character sums…”

Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators

“…The initial part of the argument is essentially a repetition with some minor modification of the standard approach, see [7,10,13], so we suppress some details. We consider the sum S I (a; N ) defined by (4) and for a sufficiently large integer K ≥ 1, we have…”