有限域$\mathbb F_{2^n}$上一类二项式的密码学性质
Cryptographic properties of a class of binomials over $\mathbb F_{2^n}$
投稿时间:2020-12-09  修订日期:2020-12-09
DOI:
中文关键词: 有限域  差分谱  Walsh谱  二次型
英文关键词: finite field  differential spectrum  Walsh spectrum  quadratic form
基金项目:国家自然科学基金项目(面上项目,重点项目,重大项目);中南民族大学中央高校基本科研业务费专项资金项目
作者单位E-mail
王一博 中南民族大学数学与统计学学院 swiftwyb@126.com 
夏永波 中南民族大学数学与统计学学院 xia@mail.scuec.edu.cn 
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中文摘要:
      研究了有限域$\mathbb F_{2^n}$上一类二次函数$F(x)=x^{2^{2t}+1}+x^{2^t+1}$的密码学性质,其中${\rm gcd}(n,t)=1$. 基于有限域上线性化多项式和二次型的理论,确定了$F(x)$的差分谱,并计算其非线性度. 特别地,当$n$为奇数时,计算出了它的Walsh谱. 最后作为应用,利用$F(x)$构造了两类线性码,并确定它们的重量分布.
英文摘要:
      The cryptographic properties of a class of binomials $F(x)=x^{2^{2t}+1}+x^{2^t+1}$ over finite field $\mathbb F_{2^n}$ are investigated where ${\rm gcd}(n,t)=1$. Based on the theory of linearized polynomials and quadratic forms, the differential spectrum of $F(x)$ is determined, and its nonlinearity is also calculated. In particular, the Walsh spectrum of $F(x)$ is obtainted for odd $n$. Finally, as applications, two binary linear codes are constructed from $F(x)$ and their weight distributions are derived.
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