Solving x~(2~k +1) + x + a = 0 in F_(2~n) with gcd(n, k) = 1

摘要

Let N-a be the number of solutions to the equation x(2k + l) + x + a = 0 in F-2(n) where gcd(k, n) = 1. In 2004, by Bluher [2] it was known that possible values of N-a are only 0, 1 and 3. In 2008, Helleseth and Kholosha [13] found criteria for N-a = 1 and an explicit expression of the unique solution when gcd(k, n) = 1. In 2010 [14], the extended version of [13], they also got criteria for N-a = 0, 3. In 2014, Bracken, Tan and Tan [5] presented another criterion for N-a = 0 when n is even and gcd(k, n) = 1. This paper completely solves this equation x(2k + l) + x + a = 0 with only the condition gcd(n, k) = 1. We explicitly calculate all possible zeros in F-2(n) of P-a (x). New criteria for which a, N-a is equal to 0, 1 or 3 are by-products of our result. (C) 2019 Elsevier Inc. All rights reserved.