9 research outputs found
Solving over Finite Fields
Solving the equation over finite field \GF{Q},
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
\GF{Q}-zeros of have been studied: in \cite{Bluher2004} it was shown
that the possible values of the number of the zeros that has in
\GF{Q} is , , or .
Some criteria for the number of the \GF{Q}-zeros of 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 , 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 \GF{Q}-zeros of are proved. For the
cases of one or two \GF{Q}-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
On a class of quadratic polynomials with no zeros and its application to APN functions
10.1016/j.ffa.2013.08.006Finite Fields and their Applications2526-3