[Resource Topic] 2019/1493: Solving $X^{q+1}+X+a=0$ over Finite Fields

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

Authors: Kwang Ho Kim, Junyop Choe, Sihem Mesnager


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.

ePrint: https://eprint.iacr.org/2019/1493

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