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Designs, Codes and Cryptography

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22-11-2022

The cycle structure of a class of permutation polynomials

Authors: Dan Zeng, Xiangyong Zeng, Lisha Li, Yunge Xu

Published in: Designs, Codes and Cryptography | Issue 4/2023

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Abstract

In this paper, we study the cycle structure of a permutation polynomial of the form \(f(x)=x^{(q+1)s_1}(x+x^{q})^{s_2}+x^{s_3}\) over \({\mathbb {F}}_{q^2}\), where \((s_1,s_2,s_3)\in \{(q^2-2,3,q), (1,q^2-2,1), (1,q^2-1,2), (1,2,4)\}\) and q is even. By calculating the sum of all elements in each cycle of a fraction polynomial \(\frac{x}{x^3+x^2+1}\) or a linearized polynomial \(x^{2^e}+x^2+x\) with \(e\ge 0\), the cycle structure of f(x) over \({\mathbb {F}}_{q^2}\) in the first three cases, that is, \((s_1,s_2,s_3)=\{(q^2-2,3,q), (1,q^2-2,1)\) or \((1,q^2-1,2)\}\), is characterized. For the case \((s_1,s_2,s_3)=(1,2,4)\), we give the cycle structure of f(x) over \({\mathbb {F}}_{q^2}\) for \(q={2^{{2}^{k}}}\) with a positive integer k. For \(q=2^{{2}^{k}_p}\) with an odd prime p, it needs more techniques to determine the cycle structure of f(x). We only give its cycle length.

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Title: The cycle structure of a class of permutation polynomials
Authors: Dan Zeng
Xiangyong Zeng
Lisha Li
Yunge Xu
Publication date: 22-11-2022
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 4/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01155-8

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