**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

**Date: **received 12 Feb 2020

**Contact author: **smesnager at univ-paris8 fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20200213:132954 (All versions of this report)

**Short URL: **ia.cr/2020/160

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