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Erschienen in: Applicable Algebra in Engineering, Communication and Computing 4/2022

03.09.2020 | Original Paper

Further results on permutation polynomials from trace functions

verfasst von: Danyao Wu, Pingzhi Yuan

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 4/2022

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Abstract

For a prime p and positive integers mn, let \({{\mathbb {F}}}_q\) be a finite field with \(q=p^m\) elements and \({{\mathbb {F}}}_{q^n}\) be an extension of \({{\mathbb {F}}}_q.\) Let h(x) be a polynomial over \({{\mathbb {F}}}_{q^n}\) satisfying the following conditions: (i) \({\mathrm{Tr}}_m^{nm}(x)\circ h(x)=\tau (x)\circ {\mathrm{Tr}}_m^{nm}(x)\); (ii) For any \(s \in {{\mathbb {F}}}_{q}\), h(x) is injective on \({\mathrm{Tr}}_m^{nm}(x)^{-1}(s),\) where \(\tau (x)\) is a polynomial over \({{\mathbb {F}}}_{q}.\) For \(b,c \in {{\mathbb {F}}}_q,\) \(\delta \in {{\mathbb {F}}}_{q^n}\), and positive integers ijd with \(q\equiv \pm 1 \pmod {d}\), we propose a class of permutation polynomials of the form
$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+c({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{j(q^n-1)}{d}}+h(x) \end{aligned}$$
over \({{\mathbb {F}}}_{q^n}\) by employing the Akbary–Ghioca–Wang (AGW) criterion in this paper. Accordingly, we also present the permutation polynomials of the form
$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+h(x) \end{aligned}$$
by letting \(c=0\) and choosing some special i, which covered some known results of this form.

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Metadaten
Titel
Further results on permutation polynomials from trace functions
verfasst von
Danyao Wu
Pingzhi Yuan
Publikationsdatum
03.09.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing / Ausgabe 4/2022
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00456-6

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