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 $$\ddot{I}ƒ(X)=X+X^d+X^{dq}\in {\mathbb{F}}_{q^2}[X].$$ For $\hat{I}\pm \in {\mathbb{F}}_q^{\ast }$, define $$f_{\hat{I}}\pm (x)={\mathrm{Tr}}_{q^2}(\hat{I}\pm (\ddot{I}{ƒ}^{-1}(x){)}^3),x\in {\mathbb{F}}_{q^2}.$$ We prove that $f_{\hat{I}}\pm$ is bent if and only if $\hat{I}\pm$ 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.

Substantially revised version. The proof of the outside Walsh coefficients has been replaced by a new self-contained Hasse-congruence and carry-graph argument. The main bentness theorem remains unchanged