Bent Functions, Spreads, and o-Polynomials

Abstract

We show that bent functions $f$ from ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$ to ${\mathbb F}_p$, which are constant or affine on the elements of a given spread of ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$, either arise from partial spread bent functions, or they are Boolean and a generalization of Dillon's class $H$. For spreads of a presemifield $S$, we show that a bent function of the second class corresponds to an o-polynomial of a presemifield in the Knuth orbit of $S$. In contrast to the finite fields case, we have to consider pairs of (pre)semifields in a Knuth orbit. We give a canonical example of an o-polynomial for commutative presemifields (which also defines a hyperoval on the semifield plane) and show that the corresponding bent functions belong to the completed Maiorana--McFarland class. Using Albert's twisted fields and Kantor's family of presemifields, we explicitly present examples of such bent functions.