发表机构
KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Hubei Province Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University（中国科学院数学与系统科学研究院; 湖北大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画线性化置换多项式的循环结构与奇偶性，为Mullen和Vaughan1988年的问题提供迪克森矩阵形式表述，推导奇偶性判据并给出判定算法，应用于编码、密码等领域。
AI 中文摘要
有限域上的线性化置换多项式在编码理论、密码学与组合数学中具有重要应用。本文基于其迪克森矩阵（Dickson matrix）的若尔当块，给出了一般线性化置换多项式循环结构的显式刻画，从而为Mullen与Vaughan于1988年提出的问题提供了迪克森矩阵形式的表述。基于该循环结构，本文建立了一个矩阵理论框架以刻画其奇偶性：对于特征为2的有限域，本文显式确定了所有奇数阶线性化置换多项式；对于奇特征域，本文推导了基于关联迪克森矩阵特征值结构的奇偶性判据，并基于该判据给出了判定给定线性化置换多项式奇偶性的算法。
英文摘要
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.
Comments20 pages
