向量布尔函数相关函数的性质分析  

On Properties of the Correlation Function of Vectorial Boolean Functions

作　　者：张婧婧 卓泽朋[1] 陈国龙 ZHANG Jingjing;ZHUO Zepeng;CHEN Guolong(School of Mathematical Sciences,Huaibei Normal University,235000,Huaibei,Anhui,China;School of Computer Engineering,Bengbu University,233030,Bengbu,Anhui,China)

机构地区：[1]淮北师范大学数学科学学院,安徽淮北235000 [2]蚌埠学院计算机工程学院,安徽蚌埠233030

出　　处：《淮北师范大学学报（自然科学版）》2023年第1期1-7,共7页Journal of Huaibei Normal University：Natural Sciences

基　　金：安徽省重点研发项目(202004a05020043);淮北师范大学研究生创新基金项目(CX2023045)。

摘　　要：文章研究向量布尔函数互相关函数的性质以及利用向量Bent函数构造Bent函数。首先,利用WalshHadamard变换以及互相关函数的性质研究向量布尔函数的互相关函数与Walsh谱之间的关系,且进一步推导出任意4个向量布尔函数的互相关函数之间的关系。其次,给出向量布尔函数的平方和指标和绝对值指标的定义,并进一步推导出平方和指标的上界以及利用分类讨论法得到其达到上界的条件。最后,利用向量布尔函数绝对值指标的性质以及Walsh-Hadamard变换,给出由向量Bent函数构造Bent函数的方法。This paper mainly studies the properties of the cross-correlation function of the vectorial Boolean function and uses the vectorial Bent function to construct the Bent function.Firstly,the Walsh-Hadamard transformation and the properties of the cross-correlation function are used to study the relationship between the cross-correlation function of the vectorial Boolean function and the Walsh spectrum,and the relationship among the cross-correlation functions of any four vectorial Boolean functions is further deduced.Secondly,the definitions of the square sum index and the absolute value index of the vectorial Boolean function are given,and the upper bound of the square sum index is further deduced and discussed by classification.Finally,based on the properties of the absolute value index of the vectorial Boolean function and the Walsh-Hadamard transformation,the method of constructing the Bent function from the vectorial Bent function is given.

关 键 词：布尔函数 向量布尔函数 自相关函数 互相关函数 BENT函数 

分 类 号：TN918.1[电子电信—通信与信息系统]