有限域上二次Bent函数的构造

doi:10.13190/jbupt.201003.52.zhangfr

北京邮电大学学报 ›› 2010, Vol. 33 ›› Issue (3): 52-56.doi: 10.13190/jbupt.201003.52.zhangfr

有限域上二次Bent函数的构造

张凤荣¹,高军涛¹,胡予濮¹,谢敏^1,2

(1. 西安电子科技大学计算机网络与信息安全教育部重点实验室，西安 710071; 
2. 广东省信息安全技术重点实验室（广州大学）, 广州 510405)

收稿日期:2009-09-07 修回日期:2009-11-20 出版日期:2010-06-28 发布日期:2010-05-14
通讯作者: 张凤荣 E-mail:zhfl203@163.com
基金资助:
国家自然科学基金项目(60833008, 60503010, 60832001)；国家重点基础研究发展计划项目(2007CB311201)

Construction of Quadratic Bent Functions over Finite Fields

ZHANG Feng-rong¹, GAO Jun-tao¹, HU Yu-pu¹, XIE Min^1,2

(1. Key Laboratory of Computer Network and Information Security, Ministry of Education, Xidian University, Xi'an 710071, China; 
2. Key Laboratory of Information Security Technology in Guangdong (Guangzhou Uni
versity), Guangzhou 510405, China)

Received:2009-09-07 Revised:2009-11-20 Online:2010-06-28 Published:2010-05-14

摘要/Abstract

摘要：

针对Ma W P等给出的二次多项式函数是Bent函数的充要条件以及Charpin P 等给出的二次三项函数是

semi-Bent函数的充要条件这2个结论进行了研究，结果表明，Ma W P等文章中的推论5、6和Charpin P

等文章中的定理5、6是不完全正确的. 结合有限域上的二次多项式理论，给出了相应的正确结果，并构

造了一类三项式和四项式Bent函数. 借助多项式置换，提出了一种利用二次二项Bent函数构造多项式Bent

函数的新方法. 

关键词: 密码函数, Boolean函数, Bent函数, 最大非线性度

Abstract:

A sufficient and necessary condition that a quadratic polynomial function is a bent function was

presented by Ma W P, et al. in 2005. In the same year, Charpin P, et al proposed a sufficient and

necessary condition that a polynomial function is a quadratic semi-Bent function with three

trace terms in a paper. Corollary 5,6 of the former and Theorem 5,6 of the latter are not quite right

by further investigating these conditions. Based on the theory of quadratic polynomial over finite

fields, the correct conditions are presented. Moreover, some quadratic bent functions with three or

four trace terms are constructed. Finally, a new method for constructing bent functions in polynomial

forms is provided by using bent functions of two trace terms and permutation of polynomial.

Key words: cryptographic functions, Boolean functions, bent functions, maximum nonlinearity

张凤荣高军涛胡予濮谢敏. 有限域上二次Bent函数的构造[J]. 北京邮电大学学报, 2010, 33(3): 52-56.

ZHANG Feng-rong, GAO Jun-tao, HU Yu-pu, XIE Min. Construction of Quadratic Bent Functions over Finite Fields[J]. JOURNAL OF BEIJING UNIVERSITY OF POSTS AND TELECOM, 2010, 33(3): 52-56.

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有限域上二次Bent函数的构造

Construction of Quadratic Bent Functions over Finite Fields

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