On a type of permutation rational functions over finite fields

摘要

Let p be a prime and n be a positive integer. Let f(b) (X) = X+ (X-p- X+ b)(-1), where b is an element of F-pn is such that Tr-pn/p(b) not equal 0. In 2008, Yuan et al. [12] showed that for p = 2, 3, f(b) permutes F-p(n) for all n = 1. Using the Hasse-Weil bound, we show that when p 3 and n = 5, f(b) does not permute F-p(n). For p 3 and n = 2, we prove that f(b) permutes F-p2 if and only if Tr-p2/p (b) = +/- 1. We conjecture that for p 3 and n = 3, 4, f(b) does not permute F-pn. (C) 2020 Elsevier Inc. All rights reserved.