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

Let $\delta \in {\mathbb{F}}_{2^n}$ satisfy ${\mathrm{Tr}}_{{\mathbb{F}}_{2^n}\mathrm{/}{\mathbb{F}}_2}(\delta )=1$. We study the permutation behavior of $$f(x)={\left(\frac{1}{x^2+x+\delta }\right)}^{2^k}+x$$ 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 $0\le k<n$, $f(x)$ is a permutation of ${\mathbb{F}}_{2^n}$ if and only if $k=0$ or $k=1$.