**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 $\gcd(k,n)=1$. 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.

**Category / Keywords: **foundations / Muller-Cohen-Matthews; polynomials Dickson polynomial; Zeros of polynomial; Irreducible polynomials

**Date: **received 18 Mar 2019

**Contact author: **smesnager at univ-paris8 fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20190320:103458 (All versions of this report)

**Short URL: **ia.cr/2019/307

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