On Dillonʼs class H of bent functions, Niho bent functions and o-polynomials

Abstract

One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana–McFarland class. We first notice that H can be extended to a slightly larger class that we denote by $\mathcal{H}$. We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander and Kholosha, are the univariate form of the functions of class $\mathcal{H}$. Their restrictions to the vector spaces $\omega {\mathbb{F}}_{2^{n/2}}$, $\omega \in {\mathbb{F}}_{2^n}^{\star }$, are linear. We also characterize the bent functions whose restrictions to the $\omega {\mathbb{F}}_{2^{n/2}}$ʼs are affine. We answer the open question raised by Dobbertin et al. (2006) in [11] on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitly calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana–McFarland class, which brings us back to the problem of knowing whether H (or $\mathcal{H}$) is a subclass of the Maiorana–McFarland completed class. We then show that the condition for a function in bivariate form to belong to class $\mathcal{H}$ is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear o-polynomials, we deduce a large number of new cases of bent functions in $\mathcal{H}$, which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana–McFarlandʼs functions).