有限域上的正规元与不可约多项式  

Normal Elements and Irreducible Polynomials in Finite Fields

作　　者：曹炜 李卫华[3] 徐碧云 Wei CAO;Wei Hua LI;Bi Yun XU(School of Mathematics and Statistics,Minnan Normal University,Zhangzhou 363000,P.R.China;Fujian Key Laboratory of Granular Computing and Applications,Zhangzhou 363000,P.R.China;School of Mathematics and Statistics,Ningbo University,Ningbo 315211,P.R.China)

机构地区：[1]闽南师范大学数学与统计学院,漳州363000 [2]福建省粒计算及其应用重点实验室,漳州363000 [3]宁波大学数学与统计学院,宁波315211

出　　处：《数学学报（中文版）》2024年第4期624-633,共10页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11871291);福建省自然科学基金资助项目(2022J02046)。

摘　　要：设F_(q)为q阶有限域,F_(q)^(n)为F_(q)的n次扩域.设α∈F_(q)^(n),若{α,α^(q),...,α^(q^(n-1))}构成F_(q)^(n)/F_(q)的一组基,称α为F_(q)^(n)/F_(q)的正规元.正规元可用来加速有限域上的算术运算,因而在编码和密码中具有重要应用.正规元的极小多项式一定是非零迹的不可约多项式,但反之未必成立.本文利用线性化多项式给出了关于该问题的一组充分必要条件,推广了已知结论.Let F_(q) be the finite field of elements,and F_(q)^(n) be its extension of degree.An element α∈F_(q)^(n) is called a normal element of F_(q)^(n)/F_(q) if {α,α^(q),...,α^(q^(n-1))}constitutes a basis of F_(q)^(n)/F_(q).Normal elements over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography.The minimal polynomial of a normal element is certainly an irreducible polynomial with nonzero trace,while the converse does not hold in general.Using linearized polynomials,we give some necessary and sufficient conditions for this problem,which extend the known results.

关 键 词：有限域 正规元 线性化多项式 不可约多项式 

分 类 号：O156[理学—数学]