Arithmetic constraints of polynomial maps through discrete logarithms

JOURNAL OF NUMBER THEORY

卷:229

Article

Reis, Lucas¹ 

[1] Univ Fed Minas Gerais, Dept Matemat, BR-30270901 Belo Horizonte, MG, Brazil

关键词: Finite fields;    Discrete logarithm;    Polynomial maps;    Cyclotomic cosets;   

DOI  :  10.1016/j.jnt.2020.10.015

来源: Elsevier

【 摘 要 】

Let q be a prime power, let F-q be the finite field with q elements and let theta be a generator of the cyclic group Fq*. For each a is an element of F-q*, let log theta a be the unique integer i is an element of{1, ... , q -1} such that a = theta i. Given polynomials P-1, . . . , P-k is an element of F-q[x] and divisors 1 < d1, . . . , dk of q- 1, we discuss the distribution of the functions Fi : y (sic) log theta Pi(y) (mod di), over the set Fq \ boolean OR ki=1{y is an element of Fq | Pi(y) = 0}. Our main result entails that, under a natural multiplicative condition on the pairs (di, Pi), the functions Fi are asymptotically independent. We also provide some applications that, in particular, relates to past work. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free