Bent partitions

Abstract

Spread and partial spread constructions are the most powerful bent function constructions. A large variety of bent functions from a 2m-dimensional vector space V ( p) 2m over F p into F p can be generated, which are constant on the sets of a partition of V ( p) 2m obtained with the subspaces of the (partial) spread. Moreover, from spreads one obtains not only bent functions between elementary abelian groups, but bent functions from V ( p) 2m to B, where B can be any abelian group of order p k , k ≤ m. As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of V(2) 2m , with these remarkable properties. In this article we present first such partitions-other than (partial) spreads-which we call bent partitions, for V ( p) 2m , p odd. We investigate general properties of bent partitions, like number and cardinality of the subsets of the partition. We show that with bent partitions we can construct bent functions from V ( p) 2m into a cyclic group Z p k . With these results, we obtain the first constructions of bent functions from V ( p) 2m into Z p k , p odd, which provably do not come from (partial) spreads.