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Cycle structures and parities of linearized permutation polynomials

  • Huajun Bian
  • Shaoshi Chen
  • Dabin Zheng
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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.

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Cite

BibTeXRIS
BibTeX
@misc{bian2026cycle,
  title = {{Cycle structures and parities of linearized permutation polynomials}},
  author = {Huajun Bian and Shaoshi Chen and Dabin Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11574},
  eprint = {2610.11574},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11574},
}
RIS
TY  - UNPB
TI  - Cycle structures and parities of linearized permutation polynomials
AU  - Huajun Bian
AU  - Shaoshi Chen
AU  - Dabin Zheng
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11574
UR  - https://arxiv.org/abs/2610.11574
AB  - 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.
ER  -