一类完全非线性函数的构造及其唯一性  

Construction and Uniqueness of a New Family of Perfect Nonlinear Functions

作　　者：周子健[1] 周悦[1] 李超[1,2] 

机构地区：[1]国防科技大学理学院数学与系统科学系,长沙410073 [2]信息保障科学技术实验室,北京100072

出　　处：《密码学报》2014年第3期279-286,共8页Journal of Cryptologic Research

基　　金：信息保障科学技术实验室开放基金项目(KJ-12-02);国家自然科学基金项目(61070215)

摘　　要：完全非线性函数和几乎完全非线性函数由于其良好的差分性质,在密码学、编码学和有限几何等众多领域有着广泛的应用.而完全非线性函数和交换半域的对应关系,使得研究有限交换半域来推动密码学、编码学等领域的发展成为可能.半域的研究始于Dickson,其后Knuth给出了半域特征的定义,使半域构造及其性质的研究成为有限几何中的热点问题.2013年,Zhou和Pott给出了一种基于Albert预半域乘法和秩为2的Cohen-Ganley预半域乘法的新的秩为2的有限预半域,本文结合Zhou和Pott预半域乘法的构造思想,参考Albert旋转预半域乘法的形式,推广提出了一类预半域并证明了参数多项式f一定是置换多项式,并且表达式是唯一确定的.结合该类预半域以及Zhou和Pott提出的一个完全非线性函数,导出了一类几乎完全非线性函数,并对其中m=4的情形,用Magma软件进行了编程测试,证明了几乎完全非线性函数是等价于另一个已知的几乎完全非线性函数,但是对于m≠4的情形,测试所得到的几乎完全非线性函数是否等价于已知的函数仍是一个值得研究的问题.Perfect nonlinear(PN) functions and almost perfect nonlinear(APN) functions have wide applications in cryptography, coding theory and finite geometry due to its good differential properties. Based on the correspondence between PN functions and commutative semifields, it is effective to study finite commutative semifields to enhance the development of cryptography and coding theory. The study of semifields was started by Dickson before Knuth giving the definition of characteristic of semifields. In 2013, Zhou and Pott came up with a family of finite presemifields(rank 2) based on multiplication of Albert presemifiled and multiplication of Conhen-Ganley presemifield(rank 2). According to the procedure of constructing the multiplication in Zhou-Pott presemifields and the form of multiplication of Albert twisted presemifield, this paper constructs a family of presemifields. This paper proves that polynomial parameter f in the presemifield is a permutation. Furthermore, we prove the uniqueness of polynomial f and induce a family of APN functions. In the situation of m =4, with the help of the software Magma, the APN function is shown to be equivalent to a known APN function, but for m ≠4, it is still an open problem to test whether the APN function constructed in this paper is equivalent to any known APN function.

关 键 词：完全非线性函数 几乎完全非线性函数 半域 有限域 置换多项式 

分 类 号：O174[理学—数学]