| 引用本文: | 曹喜望.Carlitz 定理的一个注记.[J].国防科技大学学报,2012,34(2):39-41.[点击复制] |
| CAO Xiwang.A note on a theorem of Carlitz[J].Journal of National University of Defense Technology,2012,34(2):39-41[点击复制] |
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| Carlitz 定理的一个注记 |
| 曹喜望1,2,3 |
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(1.南京航空航天大学 数学系，江苏 南京 210016;2.北京航空航天大学 数学、教育与行为教育部重点实验室，北京 100191;3.中国科学院研究生院 信息安全国家重点实验室，北京 100039)
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| 摘要: |
| 置换多项式一直是一个热门的研究课题，事实上，研究有限域上的置换多项式相当于研究有限域上的一一映射。所以它在编码密码、组合设计、代数曲线等许多领域有重要的应用。Carlitz曾经对一些置换多项式有一个刻画，〖WTBX〗证明了如果f(x)是一个系数在Fq的多项式满足f(0)=0,f(1)=1，并且对任意a,b∈Fq有η(f(a)-f(b))=η(a-b)，这里η是Fq的乘法群Fq*的二次特征，则存在某个非负整数j使得对任意x∈Fq，有f(x)=xpj.本文给出了这个结果的推广。 |
| 关键词: 有限域 置换多项式 指数和 |
| DOI： |
| 投稿日期：2011-07-28 |
| 基金项目:国家自然科学基金资助项目(10971250) |
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| A note on a theorem of Carlitz |
| CAO Xiwang1,2,3 |
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(1.Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;2.LMIB of the Ministry of Education, Beijing University of Aeronautics and Astronautics,Beijing 100191, China;3.State Key Lab of Information Security, Graduate School of Chinese Academy of Sciences, Beijing 100039, China)
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| Abstract: |
| The study of permutation polynomials over finite fields has been a hotspot research topic for a long time. In fact, it is equivalent to the study of one-to-one mapping between finite fields. Therefore, it has many important applications in coding theory, cryptography and algebraic curves, etc. Carlitz had a characterization of permutation polynomials. He proved that if f(x)is a polynomial with coefficients over finite field Fq satisfyingf(0)=f(1) and η(f(a)-f(b)=η(a-b)for every a,b∈Fq, where η is the quadratic character of Fq* . Then f(x)=xpjfor some integer. In this note, we proved that the above result is also true for any multiplicative character of Fq*. |
| Keywords: permutation polynomials finite fields exponential sums |
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