A new class of monomial bent functions

Abstract

We study the Boolean functions $f_{\lambda }:{\mathbf{F}}_{2^n}\to {\mathbf{F}}_2,n=6r$, of the form $f(x)=\mathrm{Tr}(\lambda x^d)$ with $d=2^{2r}+2^r+1$ and $\lambda \in {\mathbf{F}}_{2^n}$. Our main result is the characterization of those λ for which $f_{\lambda }$ are bent. We show also that the set of these cubic bent functions contains a subset, which with the constantly zero function forms a vector space of dimension 2r over ${\mathbf{F}}_2$. Further we determine the Walsh spectra of some related quadratic functions, the derivatives of the functions $f_{\lambda }$.