摘要
一般情况下,在布尔函数的研究中,给定一部分地址处Walsh谱值的集合A={(λ_(i),a_(i))|λ_(i)∈F_(2)^(n),a_(i)∈Z,i=0,1,2,…,m-1},寻找满足在这些地址具有给定谱值的所有n元布尔函数是很困难的。但是如果给定的地址集合是一个向量子空间,则有简单的求解方法。文章给出一种WDC算法,求解具有子空间结构地址的Walsh谱值的所有n元布尔函数以及个数。该算法包括3方面的内容:①如何构造满足这些条件的n元布尔函数;②满足这些条件的n元布尔函数有多少个?③子空间地址上的谱值满足什么条件时,才能保证满足这些条件的n元布尔函数存在。另外,Bent函数是非线性度最高的布尔函数,具有非常好的密码学性质。文章利用WDC算法并借助计算机搜索,求解出所有的6元Bent函数,共有5425430528个。
In the research of Boolean functions,there is an important problem:given a subset A={(λ_(i),a_(i))|λ_(i)∈F_(2)^(n),a_(i)∈Z,i=0,1,2,…,m-1},find all the Boolean functions f in n variables,such that f(λ_(i))=a_(i) for all i=0,1,2,…,m-1,where f(λ_(i))is the Walsh spectral value of the Boolean function f at the address λ_(i).In general,this problem is quite difficult.But if the given set of addres-ses is a vector subspace of F_(2)^(n),there is a simple solution.This paper,gives an algorithm called WDC Algorithm,that can solve this problem efficiently in this special case.The WDC Algorithm contains the following three parts:①constructs all the Boolean functions that satisfying all these conditions;②finds the number of such Boolean functions;and③gives a necessary and sufficient condition of the spectral distribution on the subspace to ensure the existence of such Boolean functions.On the other hand,the Bent function is the Boolean function that has the maximal nonlinearity,thus possess-ing excellent cryptographic properties.This paper uses WDC algorithm to find all 6-variable Bent functions by means of computer searching,the total number of such functions is 5425430528.
作者
董军武
王殊懿
曹磊
DONG Jun-wu;Wang Shu-yi;CAO Lei(School of Mathematics and Information Sciences,Guangzhou University,Guangzhou 510006,China)
出处
《广州大学学报（自然科学版）》
CAS
2024年第4期56-66,共11页
Journal of Guangzhou University:Natural Science Edition
