Cryptographic properties of a class of binomials over $\mathbb F_{2^n}$

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 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

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.

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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