半Bent函数和多输出布尔函数的构造  

Constructions of Semi-bent Functions and Multi-output Boolean Functions

作　　者：郭梦飞 孙玉娟 李路阳 GUO Meng-Fei;SUN Yu-Juan;LI Lu-Yang(State Key Laboratory of Integrated Services Networks,Xidian University,Xi’an 710071,China;State Key Laboratory of Cryptology,Beijing 100878,China;School of Telecommunication and Information Engineering,Xi’an University of Post and Telecommunications,Xi’an 710121,China)

机构地区：[1]西安电子科技大学综合业务网理论及关键技术国家重点实验室,西安710071 [2]密码科学技术国家重点实验室,北京100878 [3]西安邮电大学通信与信息工程学院,西安710121

出　　处：《密码学报》2020年第1期26-36,共11页Journal of Cryptologic Research

基　　金：国家自然科学基金(61972303,61672414);国家密码发展基金(MMJJ20170113);陕西省自然科学基础研究计划(2019JQ-867)。

摘　　要：半bent函数是一类非线性度几乎最优且平衡的布尔函数,它弥补了bent函数的一些不足,如变元个数可以是奇数,具有平衡性.半bent函数可用于对称密码系统的设计和CDMA系统中的正交可变扩频码的构造.本文利用不相交线性码构造了一类新的半bent函数,设输入维度为n,当n=2k+1时,将F2^n划分为2^k+1个[n,k]线性码和1个[n,k+1]线性码,通过从该码集中选取合适线性码作支撑集来构造新的半bent函数.另一方面,多输出布尔函数(向量值函数)在应用中的效率更高,因此其使用场景更为广泛.本文同时利用不相交线性码构造了(n,n-k)平衡的多输出布尔函数,其中n/3<k<n/2.在保证高非线性度的条件下,其输出变量维数大于输入变量维数的一半.Semi-bent functions are balanced Boolean functions with high nonlinearity,which complements some shortcomings of bent functions.For example,semi-bent functions are balanced and the number of input variables can be odd.Apart from the applications in cryptography,semi-bent functions are also widely used in certain combinatorial designs such as construction of orthogonal variable spreading factor codes used in CDMA systems and for the sequence design.In this paper,a new construction method of semi-bent functions by using disjoint linear codes is proposed.For example,when n=2k+1,F2^n can be divided into 2^k+1 of[n,k]linear codes and one[n,k+1]linear code,these disjoint linear codes can be used to construct new semi-bent functions.In addition,multi-output Boolean functions are very efficient in many applications,so they are widely used.This paper constructs a class of(n,n-k)balanced multi-output Boolean functions by using disjoint linear codes,where n/3<k<n/2.Under the condition of high nonlinearity,the number of output variables of the functions is greater than half of the number of input variables.

关 键 词：布尔函数 非线性度 半bent 不相交线性码 向量值函数 

分 类 号：TP309.7[自动化与计算机技术—计算机系统结构]