Welcome to the resource topic for 2019/307
Title:
Solving x^{2^k+1}+x+a=0 in \mathbb{F}_{2^n} with \gcd(n,k)=1
Authors: Kwang Ho Kim, Sihem Mesnager
Abstract:Let N_a be the number of solutions to the equation x^{2^k+1}+x+a=0 in \mathbb F_{n} where \gcd(k,n)=1. In 2004, by Bluher it was known that possible values of N_a are only 0, 1 and 3. In 2008, Helleseth and Kholosha have got criteria for N_a=1 and an explicit expression of the unique solution when \gcd(k,n)=1. In 2014, Bracken, Tan and Tan presented a criterion for N_a=0 when n is even and \gcd(k,n)=1. This paper completely solves this equation x^{2^k+1}+x+a=0 with only condition \gcd(n,k)=1. We explicitly calculate all possible zeros in \mathbb F_{n} of P_a(x). New criterion for which a, N_a is equal to 0, 1 or 3 is a by-product of our result.
ePrint: https://eprint.iacr.org/2019/307
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