On the Equation $x^{2^l+1}+x+a=0$ over $\mathrm{GF}(2^k)$ (Extended Version)

TL;DR

This paper investigates the zeros of specific polynomials over finite fields, providing new criteria for their number of solutions and explicit formulas, especially when certain gcd conditions are met.

Contribution

It introduces new criteria for the number of zeros of the polynomial $P_a(x)$ over GF(2^k), including a condition for exactly one zero when gcd(l,k)=1, using permutation polynomials.

Findings

Criteria for the number of zeros of $P_a(x)$ in GF(2^k)

Explicit formulas for zeros of related polynomials

A new condition for a single zero when gcd(l,k)=1

Abstract

In this paper, the polynomials $P_a(x)=x^{2^l+1}+x+a$ with $a\in\mathrm{GF}(2^k)$ are studied. New criteria for the number of zeros of $P_a(x)$ in $\mathrm{GF}(2^k)$ are proved. In particular, a criterion for $P_a(x)$ to have exactly one zero in $\mathrm{GF}(2^k)$ when $\gcd(l,k)=1$ is formulated in terms of the values of permutation polynomials introduced by Dobbertin. We also study the affine polynomial $a^{2^l}x^{2^{2l}}+x^{2^l}+ax+1$ which is closely related to $P_a(x)$. In many cases, explicit expressions for calculating zeros of these polynomials are provided.