## Cryptology ePrint Archive: Report 2020/160

Solving Some Affine Equations over Finite Fields

Sihem Mesnager and Kwang Ho Kim and Jong Hyok Choe and Dok Nam Lee

Abstract: Let $l$ and $k$ be two integers such that $l|k$. Define $T_l^k(X):=X+X^{p^l}+\cdots+X^{p^{l(k/l-2)}}+X^{p^{l(k/l-1)}}$ and $S_l^k(X):=X-X^{p^l}+\cdots+(-1)^{(k/l-1)}X^{p^{l(k/l-1)}}$, 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.

Category / Keywords: foundations / Affine equation, Finite field, Zeros of a polynomial, Linearized polynomial