Niho型幂函数差分谱的完整刻画
Complete characterization of the differential spectrum of a Niho type power function
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中文总结 AI 辅助
本文研究有限域上Niho型幂函数的差分谱，通过有限域方程解数分析，确定其局部差分均匀性，得到新的无穷族局部差分4-均匀幂函数。
中文摘要 AI 辅助
具有Niho指数的幂函数因其在序列设计、编码理论和密码学中的重要应用而受到广泛关注。本文研究有限域$\boldsymbol{\text{F}}_{2^{2m}}$上形如$F(x)=x^{s(2^m-1)+1}$（其中$2 \leq s \leq 2^m$）的Niho型幂函数的差分性质。我们首先通过Walsh谱建立$F(x)$的差分谱至多含三个非零值的一般刻画；接着针对$s=(2^k+1)^{-1} \bmod (2^m+1)$且$\text{gcd}(k,m)=e$的情形，对有限域上某些方程的解数进行精细化分析，具体证明：当$\text{gcd}(2^k-1,2^m+1)=2^e+1$时，$F(x)$是局部差分$2^e$-均匀的；当$\text{gcd}(2^k-1,2^m+1)=1$时，$F(x)$是局部差分$(2^{2e}-2^e)$-均匀的，并完全确定了这两种情形下的差分谱。这些结果完整刻画了该族函数的差分性质，并得到了新的无穷族局部差分4-均匀幂函数。
英文摘要
Power functions with Niho exponents have attracted considerable attention due to their important applications in sequence design, coding theory, and cryptography. This paper investigates the differential properties of Niho type power functions of the form $F(x)=x^{s(2^m-1)+1}$ over $\mathbb{F}_{2^{2m}}$ with $2\leq s\leq 2^m$. We first establish a general characterization of the differential spectrum of $F(x)$ having at most three nonzero values via its Walsh spectrum. Focusing subsequently on the case $s=(2^k+1)^{-1} \pmod{2^m+1}$ where $\gcd(k,m)=e$, we employ a refined analysis of the number of solutions to certain equations over finite fields. Specifically, it is proved that $F(x)$ is locally differentially $2^e$-uniform when $\gcd(2^k-1,2^m+1)=2^e+1$ and locally differentially $(2^{2e}-2^e)$-uniform when $\gcd(2^k-1,2^m+1)=1$, and their differential spectra are completely determined. These results completely characterize the differential properties of this family and yield new infinite families of locally differentially $4$-uniform power functions.
