Exponential sums with Dickson polynomials
type: Item
instance of: scholarly article
title: Exponential sums with Dickson polynomials
zbMATH Open document ID: 1104.11055
author: Jaime Gutierrez
author: Igor E. Shparlinski
author: Domingo Gómez-Pérez
published in: Finite Fields and their Applications
publication date: 2006-02-07T00:00:00Z
review text: Given a polynomial \(F(X)\in F_p[X]\) of degree at least \(2\), the \textit{nonlinear congruential generator} \((x_n)\) of elements of \(F_p\) is defined by the recurrence relation \[ x_{n+1} \equiv F(x_n) \pmod p, \qquad n = 0, 1, \dots, \] with some initial value \(x_0\). In \textit{H. Niederreiter} and \textit{I. E. Shparlinski} [Finite Fields Appl. 5, No.~3, 246--253 (1999; Zbl 0942.11037)] a new method has been invented to estimate exponential sums with such sequences for arbitrary polynomials \(F(X)\) and thus to study the distribution of the sequence. Unfortunately, for general polynomials, this method leads to rather weak bounds. However, in the special case of the \textit{inversive congruential generator} this method leads to a much stronger bound, see \textit{H. Niederreiter} and \textit{I. E. Shparlinski} [Math. Comput. 70, No.~236, 1569--1574 (2001; Zbl 0983.11048)]. For another special class of polynomials, namely for \textit{monomials} an alternative approach, producing much stronger bounds has been proposed in [\textit{J. B. Friedlander, J. S. Hansen} and \textit{I. E. Shparlinski}, Nathanson, Melvyn B. (ed.), Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, DIMACS. Series in Discrete Mathematics and Theoretical Computer Science 64, 71--79 (2004; Zbl 1086.11039)] and \textit{J. B. Friedlander} and \textit{I. E. Shparlinski} [Math. Comput. 70, No.~236, 1575--1589 (2001; Zbl 1029.11042)]. This article deals with another special case of the nonlinear congruential pseudorandom number generator constructed via \textit{Dickson polynomials}. The authors apply the method for the power generator mentioned above. However, the results are weaker since Dickson polynomials don't have several specific properties of monomials. However, in an important special case, using a bound from \textit{W.-C. Li} [Finite Fields Appl. 12, No.~1, 1--15 (2006; Zbl 1103.11025)], they obtain a result of the sa…
reviewed by: Arne Winterhof
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 5004383
zbMATH Keywords: exponential sums
zbMATH Keywords: Dickson polynomials
zbMATH Keywords: nonlinear pseudorandom number generators
MaRDI profile type: Publication
full work available at URL: j.ffa.2004.08.001
OpenAlex ID: W2136767493
cites work: Polynomial Gauss sums
cites work: Character sums with exponential functions
cites work: On the distribution of the power generator
cites work: Q4325515
cites work: On the distribution and lattice structure of nonlinear congruential pseudorandom numbers
cites work: Q4549492
cites work: Q4426679
DOI: 10.1016/J.FFA.2004.08.001
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description: scientific article; zbMATH DE number 5004383