一类Bent函数的二阶非线性度下界  

The Lower Bound on the Second-Order Nonlinearity for a Class of Bent Functions

作　　者：李春雷[1,2] 张焕国[1,2] 曾祥勇[3,4] 胡磊[4] 

机构地区：[1]武汉大学计算机学院,武汉430079 [2]空天信息安全与可信计算教育部重点实验室,武汉430079 [3]湖北大学数学与计算机科学学院,武汉430062 [4]中国科学院研究生院信息安全国家重点实验室,北京100049

出　　处：《计算机学报》2012年第8期1588-1593,共6页Chinese Journal of Computers

基　　金：国家自然科学基金(60970115;60970116;61003267;91018008;61003268)资助~~

摘　　要：为了防止存在有效的低次函数逼近,对于较小的正整数r,用于对称密码系统中的布尔函数应具有较高的r-阶非线性度.当r>1时,准确计算布尔函数的r-阶非线性度十分困难,已有的研究工作主要是通过分析其导函数的(r-1)-阶非线性度来确定布尔函数的r-阶非线性度下界.对于整数n≡2(mod 4),文中确定了一类由Niho指数生成的Bent函数的二阶非线性度下界.与相同变元个数的两类Bent函数和三类布尔函数相比,这类Bent函数具有更紧的二阶非线性度下界.The Boolean functions used in symmetric ciphers should have high rth-order nonlinearity to resist against the low-degree approximation cryptanalysis. For an integer r〉1, it is quite difficult to compute the rth-order nonlinearity of a Boolean function, and the known literatures mainly utilize the （r-1）th-order nonlinearity of its derivatives to deduce the lower bound on the rtb-order nonlinearity. For an integer n≡ 2 （mod4）, this paper investigates the nonlinearities of the corresponding derivative functions for a class of Bent functions constructed from Niho exponents, and then determines the lower bound of the second-order nonlinearity for this class of Bent functions. Compared with two Bent functions and three classes of Boolean functions with the same number of variables, this class of Bent functions has a tighter lower bound on the second-order nonlinearity.

关 键 词：BENT函数 二阶非线性度 双线性函数 WALSH谱 REED-MULLER码 

分 类 号：TP309[自动化与计算机技术—计算机系统结构]