| 摘要: |
| 有限域上的正规基在编码理论、密码学等领域有广泛的应用,是有限域研究的重要内容之一；设素数p为有限域Fq的特征,n(≥2)是正整数, ξ是Fqn在Fq上的正规元； 满足某种特殊条件的正规元的存在性一直是正规基研究的热点之一，特征和方法通常是研究有限域上特殊元素存在性的有力工具； 利用特征和估计给出了ξ和ξ+ξ－1同时为Fqn在Fq上的正规元的一个充分条件, 并由此得到了几种情形下q,n的下界， 特别地, 当n=p=2时, 给出了ξ的准确计数公式。 |
| 关键词: 有限域, 正规基, 特征和 |
| DOI： |
| 分类号: |
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| The Existence of One Class of Special Normal Bases over Finite Fields |
|
LI Bo
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| Abstract: |
| Normal bases are used in coding theory, cryptography and other fields widely, and they play an important part in the study of finite fields. Let the prime p be the characteristic of the finite field Fq, n(≥2) be an integer and ξ a normal bases generator of Fqn over Fq. The existence of normal elements that satisfy some special conditions has always been one of the hot spots in the research about normal bases. Usually, Character sums are powerful tools for studying the existence of special elements in finite fields. We give a sufficient condition such that ξ and ξ+ξ-1 are normal bases generator of Fqn over Fq by character sums estimate, and obtain some lower bounds of q and n from the sufficient condition. When n=p=2, we give a exact counting formula of ξ satisfying the above-mentioned condition. |
| Key words: finite fields, normal bases, character sums |
