两类幂函数差分谱及应用  

Differential Spectrum of Two Classes of Power Functions and Applications

作　　者：周慧娟 卓泽朋[1] 陈国龙 ZHOU Huijuan;ZHUO Zepeng;CHEN Guoong(School of Mathematics and Statistics,Huaibei Normal University,235000,Huaibei,Anhui,China;School of Computer Engineering,Bengbu University,233030,Bengbu,Anhui,China)

机构地区：[1]淮北师范大学数学与统计学院,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)》2025年第1期1-5,共5页Journal of Huaibei Normal University:Natural Sciences

基　　金：安徽省高校科学研究计划项目(2022AH050374);淮北师范大学结余经费资助项目(2023ZK032,RE230438)。

摘　　要：研究有限域上两类幂函数f(x)=x^(2)和f(x)=x^(3)差分谱以及幂函数f(x)=x^(3)性质,旨在探究构造新的几乎完全非线性函数方法。利用相关差分方程解个数,刻画有限域上两类幂函数f(x)=x^(2)和f(x)=x^(3)差分谱,并通过代数学软件Magma验证理论证明结果正确性。根据Walsh谱和非线性度之间关系得到幂函数f(x)=x^(3)Walsh谱上界和非线性度下界。通过计算确定幂函数f(x)=x^(3)2个密码学指标ambiguity和deficiency取值。研究推动几乎完全非线性函数研究和应用。The differential spectrum and the properties of power functions of two types of power functions f(x)=x^(2) and f(x)=x^(3) over finite fields have been computed in order to explore methods for constructing new Almost Perfect Nonlinear Functions.By utilizing the number of solutions to relevant differential equations,the differential spectrum of two types of power functions f(x)=x^(2) and f(x)=x^(3) over finite fields are characterized.And the correctness of the theoretical proof results is verified through the algebraic software Magma.The upper bound of the Walsh spectrum and the lower bound of the Nonlinearity of power functions are obtained based on the relationship between the Walsh spectrum and nonlinearity.The values of the two cryptographic indicators of the power function f(x)=x^(3),i.e.ambiguity and deficiency,are determined by calculation.These studies have promoted the research and application of almost perfect nonlinear functions.

关 键 词：有限域 幂函数 差分谱 密码函数 

分 类 号：TN918.1[电子电信—通信与信息系统]