**Solving $X^{q+1}+X+a=0$ over Finite Fields**

*Kwang Ho Kim and Junyop Choe and Sihem Mesnager*

**Abstract: **Solving the equation $P_a(X):=X^{q+1}+X+a=0$ over finite field
$\GF{Q}$, where $Q=p^n, q=p^k$ and $p$ is a prime, arises in many
different contexts including finite geometry, the inverse Galois
problem \cite{ACZ2000}, the construction of difference sets with
Singer parameters \cite{DD2004}, determining cross-correlation
between $m$-sequences \cite{DOBBERTIN2006,HELLESETH2008} and to
construct error-correcting codes \cite{Bracken2009}, as well as to
speed up the index calculus method for computing discrete logarithms
on finite fields \cite{GGGZ2013,GGGZ2013+} and on algebraic curves
\cite{M2014}.

Subsequently, in \cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019}, the $\GF{Q}$-zeros of $P_a(X)$ have been studied: in \cite{Bluher2004} it was shown that the possible values of the number of the zeros that $P_a(X)$ has in $\GF{Q}$ is $0$, $1$, $2$ or $p^{\gcd(n, k)}+1$. Some criteria for the number of the $\GF{Q}$-zeros of $P_a(x)$ were found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is to identify all the $\GF{Q}$-zeros, even in the case $p=2$, it was solved only under the condition $\gcd(n, k)=1$ \cite{KM2019}.

We discuss this equation without any restriction on $p$ and $\gcd(n,k)$. New criteria for the number of the $\GF{Q}$-zeros of $P_a(x)$ are proved. For the cases of one or two $\GF{Q}$-zeros, we provide explicit expressions for these rational zeros in terms of $a$. For the case of $p^{\gcd(n, k)}+1$ rational zeros, we provide a parametrization of such $a$'s and express the $p^{\gcd(n, k)}+1$ rational zeros by using that parametrization.

**Category / Keywords: **foundations / Equation $\cdot$ M\"{u}ller-Cohen-Matthews (MCM) polynomial $\cdot$ Dickson polynomial $\cdot$ Zeros of a polynomial $\cdot$ Irreducible polynomial.

**Date: **received 29 Dec 2019

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

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

**Version: **20191230:193850 (All versions of this report)

**Short URL: **ia.cr/2019/1493

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