Paper 2019/307

Solving $x^{2^k+1}+x+a=0$ in ${\mathbb{F}}_{2^n}$ with $gcd(n,k)=1$

Kwang Ho Kim and 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 $$. In 2014, Bracken, Tan and Tan presented a criterion for $$ when $$ is even and $$. This paper completely solves this equation $$ with only condition $$. We explicitly calculate all possible zeros in $$ of $$. New criterion for which $$, $$ is equal to $$, $$ or $$ is a by-product of our result.