Paper 2015/168

More PS and H-like bent functions

C. Carlet

Abstract

Two general classes (constructions) of bent functions are derived from the notion of spread. The first class, $\mathcal{P}\mathcal{S}$, gives a useful framework for designing bent functions which are constant (except maybe at 0) on each of the $m$-dimensional subspaces of ${\mathbb{F}}_{2^{2m}}$ belonging to a partial spread. Explicit expressions (which may be used for applications) of bent functions by means of the trace can be derived for subclasses corresponding to some partial spreads, for instance the ${\mathcal{P}\mathcal{S}}_{ap}$ class. Many more can be. The second general class, $H$, later slightly modified into a class called $\mathcal{H}$ so as to relate it to the so-called Niho bent functions, is (up to addition of affine functions) the set of bent functions whose restrictions to the subspaces of the Desarguesian spread (the spread of all multiplicative cosets of $$, added with 0, in $$) are linear. It has been observed that the functions in $$ are related to o-polynomials, and this has led to several classes of bent functions in bivariate trace form. In this paper, after briefly looking at the $$ functions related to the André spreads, and giving the trace representation of the $$ corresponding bent functions and of their duals, we show that it is easy to characterize those bent functions whose restrictions to the subspaces of a spread are linear, but that it leads to a notion extending that of o-polynomial, for which it seems a hard task to find examples. We illustrate this with the André spreads and also study three other cases of $$-like functions (related to other spreads).