Paper 2019/1493
Solving over Finite Fields
Kwang Ho Kim, Junyop Choe, and Sihem Mesnager
Abstract
Solving the equation over finite field
, where and 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 -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 -zeros of have been studied: in
\cite{Bluher2004} it was shown that the possible values of the
number of
the zeros that has in is , , or .
Some criteria for the number of the -zeros of were
found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}.
However, while the ultimate goal is to identify all the
-zeros,
even in the case , it was solved only under the condition \cite{KM2019}.
We discuss this equation without any restriction on and
. New criteria for the number of the -zeros of
are proved. For the cases of one or two -zeros, we
provide explicit expressions for these rational zeros in terms of
. For the case of rational zeros, we provide a
parametrization of such 's and express the
rational zeros by using that parametrization.