Resolving a conjecture on permutation polynomials over $\mathbb{F}_{2^n}$

Source abstract

Let $δ\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(δ)=1$. We study the permutation behavior of $$f(x) = \left(\frac{1}{x^2+x+δ}\right)^{2^k}+x$$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerical evidence suggests that no other cases occur. In this paper, we confirm their assertion by proving that, for $0\leq k<n$, $f(x)$ is a permutation of $\mathbb{F}_{2^n}$ if and only if $k=0$ or $k=1$.