A proof of a permutation-inverse bent-function conjecture

Abstract

Let $q=2^e$ with $e$ even, and let $\mathbb F_{q^2}$ be the finite field of order $q^2$. Put $d=(q^2+q+1)/3$, and consider the permutation polynomial $$ σ(X)=X+X^d+X^{dq}\in\mathbb F_{q^2}[X].$$ For $α\in\mathbb F_q^*$, define $$f_α(x)=\operatorname{Tr}_{q^2}\bigl(α(σ^{-1}(x))^3\bigr),\qquad x\in\mathbb F_{q^2}.$$ We prove that $f_α$ is bent if and only if $α$ is not a cube in $\mathbb F_q$, thereby proving a conjecture of Li, Li, Helleseth, and Qu. The proof computes the Walsh values on $\mathbb F_q$ directly and treats the complementary parameters by reducing them to a two-variable exponential sum. A binary Hasse congruence, proved through finite carry analysis and a projective-frame cancellation of the only large carry component, forces the outside Walsh coefficients in the noncubic case to be $\pm q$. As an application, we identify a recent cyclotomic family of Xie, Li, Wang, and Zeng with the same construction in different coordinates and thereby prove their conjecture.