Solving [formula omitted] in [formula omitted] with [formula omitted

Abstract

Let Na be the number of solutions to the equation x2k+1+x+a=0 in F2n where gcd⁡(k,n)=1. In 2004, by Bluher [2] it was known that possible values of Na are only 0, 1 and 3. In 2008, Helleseth and Kholosha [13] found criteria for Na=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 Na=0,3. In 2014, Bracken, Tan and Tan [5] presented another criterion for Na=0 when n is even and gcd⁡(k,n)=1.This paper completely solves this equation x2k+1+x+a=0 with only the condition gcd⁡(n,k)=1. We explicitly calculate all possible zeros in F2n of Pa(x). New criteria for which a, Na is equal to 0, 1 or 3 are by-products of our result.