一类代数免疫度最优的奇数变元旋转对称布尔函数的构造  

Construction of a Class of Odd-variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity

作　　者：沈黎鹏 陈克非[1,2,3] SHEN Li-Peng;CHEN Kei-Fei(College of Science,Hangzhou Normal University,Hangzhou 311121,China;Hangzhou Key Laboratory of Cryptography and Network Security,Hangzhou 311121,China;Weststone Cryptologic Research Center,Beijing 100190,China)

机构地区：[1]杭州师范大学理学院,杭州311121 [2]杭州市密码与网络安全重点实验室,杭州311121 [3]卫士通摩石实验室,北京100070

出　　处：《密码学报》2019年第4期526-540,共15页Journal of Cryptologic Research

基　　金：国家自然科学基金(61472114)~~

摘　　要：密码函数包含布尔函数与向量布尔函数两大类,其密码学性质关系到整个密码系统的安全性.旋转对称布尔函数是一类输出值在输入的循环移位下保持不变的布尔函数,具有结构简单、资源利用率高、运算速度快等优点,在分组密码S盒和Hash函数的设计中有着广泛应用.本文基于正整数拆分理论,构造了一类奇变元的旋转对称布尔函数.新构造的n元布尔函数不但代数免疫度达到了最优,而且在n≥25时的非线性度是目前同类构造中最高的.此外,还证明了此类函数具有最优的代数次数,如果n≠2^m+1,m≥3.研究结果表明,构造的布尔函数具有优良的密码学性质,这对构造理论的创新和实际布尔函数的选择有着重要的意义.Boolean functions and vector Boolean functions are two large classes of cryptographic functions, and their cryptographic properties are related to the security of cryptosystems. The rotation symmetric Boolean functions is a class of Boolean functions whose output values remain invariant under circular permutations of their variables. They are widely applied to the design of symmetric cryptosystems and Hash functions because of the advantages of simple structure, high resource utilization rate and fast operation speed. Based on the theory of positive integer splitting, this paper constructs a class of odd-variable rotation symmetric Boolean functions. The newly constructed nvariable functions achieve the optimal algebraic immunity, and have the best nonlinearity in other similar constructions when n≥25. In addition, it is proved that such functions have the optimal algebraic degree, if n =≠2^m+ 1, m≥3. The results show that the constructed Boolean functions have good cryptographic properties, which is of great significance to the innovation of construction theory and the selection of practical Boolean functions.

关 键 词：代数免疫度 旋转对称布尔函数 非线性度 代数次数 

分 类 号：TP309.7[自动化与计算机技术—计算机系统结构]