Year: 2026
Type: preprint
Abstract: Linearized permutation polynomials over finite fields have important applications in coding theory, cryptography, and combinatorics. In this paper, we give an explicit description of the cycle structure of a general linearized permutation polynomial in terms of the Jordan blocks of its Dickson matrix, thereby providing a Dickson-matrix formulation of a question raised by Mullen and Vaughan in 1988. Building on the cycle structure, we establish a matrix-theoretic framework to characterize their parity. For finite fields of characteristic 2, we determine all odd linearized permutation polynomials explicitly. For fields of odd characteristic, we derive parity criteria in terms of the eigenvalue structure of the associated Dickson matrix. Based on the parity criteria, we then give an algorithm for deciding the parity of a given linearized permutation polynomial.
Source: arXiv (Cornell University)
Authors: Bian, Huajun, Chen, Shaoshi, Zheng, Dabin
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3220373537, G3894628322, G8996357861
OpenAlex ID: W7221096862