一类二值差分函数的密码学性质及其应用  

On the cryptographic properties of a class of differentially two-valued functions and their applications

作　　者：杜俊雨 夏永波[1] DU Junyu;XIA Yongbo(College of Mathematics and Statistics,South-Central University for Nationalities,Wuhan 430074,China)

机构地区：[1]中南民族大学数学与统计学学院,武汉430074

出　　处：《中南民族大学学报（自然科学版）》2022年第1期109-115,共7页Journal of South-Central University for Nationalities：Natural Science Edition

基　　金：国家自然科学基金资助项目(61771021,62171479);中央高校基本科研业务费专项资金资助项目(CZT20023);中南民族大学研究生学术创新基金资助项目(3212021sycxjj316)。

摘　　要：研究了有限域F_(2^(n))上回旋镖均匀度为4的密码函数F(x)=x^(2^(t)+2)+γx的性质,其中n≡2(mod4),t=n/2且ord(γ^(2^(t)-1))=3.通过研究差分方程解的数目,确定了F(x)的差分谱,结果表明该函数是二值差分的.利用二次型理论,计算了F(x)的Walsh谱,进而确定了其非线性度.作为应用,利用F(x)构造一类二元线性码,确定了该线性码的重量分布.最后通过二值差分函数和2-设计之间的关系,利用该函数构造了一个2-设计.The cryptographic properties of the function F(x)=x^(2^(t)+2)+γx with 4-uniform BCT over F_(2^(n)) are investigated,where n≡2(mod4),t=n/2 and ord(γ^(2^(t)-1))=3.By studying the number of solutions of derivative equation,the differential spectrum of F(x)is determined,and the result shows that F(x)is differentially two-valued.Utilizing the theory of quadratic forms,the Walsh spectrum of F(x)is also calculated,so its nonlinearity is determined.As applications,a class of binary linear codes is constructed from F(x),and the weight distributions of these linear codes are derived.Moreover,based on the relationship between differentially two-valued functions and 2-designs,a 2-design is also proposed.

关 键 词：差分谱 二值差分函数 线性码 重量分布 2-设计 

分 类 号：O157[理学—数学]