Hyperovals and bent functions

Abstract

We consider Niho bent functions (they are equivalent to bent functions which are linear on the elements of a Desarguesian spread). Niho bent functions are in one-to-one correspondence with line ovals in an affine plane. Furthermore, Niho bent functions are in one-to-one correspondence with ovals (in a projective plane) with nucleus at a designated point. We introduce a new analog of o-polynomials for description of hyperovals and present a criteria for existence of such polynomials. These connections allow us to present new simple descriptions of the Adelaide and Subiaco hyperovals and their automorphism groups. There are notions of duality for bent functions and for projective planes, we show that these notions are consistent for Niho bent functions.