Near完美非线性函数及有关课题研究

Nonlinear functions are one of the most important topics in combinatorics. Functions with high nonlinearity have important applications in cryptography, sequences,coding theory and other research objects in combinatorics. During the last 20 years, there has been a lot of studies of nonlinear functions with high nonlinearity. Let f be a function from an abelian group (A,+) to another abelian group (B,+), the past research of nonlinear function mainly focus on the optimal nonlinearity of f in the case the order of A is divisible by the order of B and f is called a perfect nonlinear function if the nonlinearity of f meets optimality; but the optimal nonlinearity of f consider less for the order of A is not divisible by the order of B.We makes an investigation into the nonlinearity of a function f in the case the order of A is not divisible by the order of B and the notion of a near perfect nonlinear function is then proposed. The project mainly study the constrution of near perfect nonlinear function and the related applications of nonlinear functions.

非线性函数是组合数学中的重要研究对象。非线性度高的函数在密码学，序列理论，编码理论，以及组合数学中的其他对象的研究中有广泛的应用。过去的二十年里，由于其众多的应用，非线性函数被大量的研究。设A，B是有限交换群，运算为加法，用f表示A到B的函数，过去研究的非线性函数主要考虑B的阶整除A的阶的条件下函数f的非线性最优性并把达到非线性最优性的函数f称为完美非线性函数；但对于B的阶不整除A的阶情况下函数的非线性最优性，文献中结果很少。我们此前已较为深入地研究了B的阶不整除A的阶情况下函数的非线性最优性并把达到非线性最优的函数称为Near完美非线性函数，从而提出了Near完美非线性函数的概念并把对非线性函数的研究引入了一个新的方向。本项目主要研究Near完美非线性函数的构造及非线性函数相关的应用。

本项目按计划进行并达到了预期目标。本项目中研究了Near完美非线性函数的构造；研究了非线性函数相关的应用：(1)利用非线性度高的函数构造新的最优跳频序列集；(2)利用ZD非线性函数构造最优差系统集并得到新的最优差系统集。本项目共在国外期刊发表了5篇期刊论文，其中在SCI期刊上发表1篇论文。