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arXiv · 2507.00781

Permutation polynomials of the form $x+\gamma \mathrm{Tr}(H(x))$

Yangcheng Li·Xuan Pang·Pingzhi Yuan·Yuanpeng Zeng
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Abstract

Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + \gamma \mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.\] We present some properties of the set \(P_H\), particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set \(P_H\) and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set \(P_H\) reaches this upper bound, the function \(\mathrm{Tr}(H(x))\) must be an \(\mathbb{F}_q\)-linear function. Finally, we study two classes of functions $H(x)$ over \(\mathbb{F}_{q^2}\) and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case.

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Yangcheng Li, Xuan Pang, Pingzhi Yuan, Yuanpeng Zeng. 2025-07-01. Permutation polynomials of the form $x+\gamma \mathrm{Tr}(H(x))$. https://arxiv.org/abs/2507.00781

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