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**2020/160**

**Title:**

Solving Some Affine Equations over Finite Fields

**Authors:**
Sihem Mesnager, Kwang Ho Kim, Jong Hyok Choe, 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.

**ePrint:**
https://eprint.iacr.org/2020/160

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