The complete cubic Walsh spectrum of a permutation-inverse Boolean family

Abstract

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.