Solving Some Affine Equations over Finite Fields

Abstract: Let $l$ and $k$ be two integers such that $l|k$. Define $T_l^{k(X):=X+X{pl}+\cdots+X{p{l(k/l-2)}}+X{p{l(k/l-1)}}$} and $S_l^{k(X):=X-X{pl}+\cdots+(-1){(k/l-1)}X{p{l(k/l-1)}}$,} where $p$ is any prime. This paper gives explicit representations of all solutions in $\GF{p^n}$ to the affine equations $T_l^{k}(X)=a$ and $S_l^{k}(X)=a$, $a\in \GF{p^n}$. For the case $p=2$ that was solved very recently in \cite{MKCL2019}, the result of this paper reveals another solution.