AI 中文总结
研究置换逆布尔族的完全三次沃尔什谱，在\(\alpha\in(\mathbb{F}_q^*)^3\)情况下，利用外部沃尔什系数完备化方法等，确定相关布尔函数的沃尔什分布，其值为\(0\)和\(\pm 2q\)，还借助APN性质等确定逐点三次谱。
AI 中文摘要
设\(q = 2^e\)，\(e\geq2\)为偶数，\(d=(q^2 + q + 1)/3\)，\(\sigma(X)=X + X^d + X^{dq}\)是由丁、曲、王、袁、袁引入的\(\mathbb{F}_{q^2}\)的置换。对于\(\alpha\in\mathbb{F}_q^*\)，定义布尔函数\(f_\alpha(x)=\text{Tr}_{q^2}(\alpha(\sigma^{-1}(x))^3)\)，\(x\in\mathbb{F}_{q^2}\)。本文确定了\(\alpha\in(\mathbb{F}_q^*)^3\)剩余三次情况下\(f_\alpha\)的完全沃尔什分布。这些函数不是弯曲的，而是\(2 -\)平台的：其沃尔什值恰好为\(0\)和\(\pm 2q\)，且具有精确的重数。主要新工具是外部沃尔什系数的完备化方法：由外部约化产生的穿孔傅里叶变换在缺失线上被填充，对外部频率不可见的修改，然后将完备化后的函数与Kasami APN单项式的布尔分量进行识别。APN性质提供了一个四阶矩恒等式，它与已知的子域谱和哈塞可除性同余一起，确定了逐点三次谱。
英文摘要
Let $q=2^e$ with $e\ge2$ even, put $d=(q^2+q+1)/3$, and let $σ(X)=X+X^d+X^{dq}$ be the permutation of $\mathbb F_{q^2}$ introduced by Ding, Qu, Wang, Yuan, and Yuan. For $α\in\mathbb F_q^*$, define the Boolean function \[ f_α(x)=\operatorname{Tr}_{q^2}\bigl(α(σ^{-1}(x))^3\bigr), \qquad x\in\mathbb F_{q^2}. \] In this paper, we determine the complete Walsh distribution of $f_α$ in the remaining cubic case $α\in(\mathbb F_q^*)^3$. More precisely, these functions are not bent but are $2$-plateaued: their Walsh values are precisely $0$ and $\pm 2q$, with exact multiplicities. The main new tool is a completion method for the outside Walsh coefficients: the punctured Fourier transform arising from the outside reduction is filled on the missing line, a modification invisible to outside frequencies, and the completed function is then identified with a Boolean component of a Kasami APN monomial. The APN property supplies a fourth-moment identity which, together with the known subfield spectrum and a Hasse divisibility congruence, forces the pointwise cubic spectrum.
Comments15 pages
