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 ${\mathbb {V}}_{2m}^{(p)}$ over ${\mathbb {F}}_p$ into ${\mathbb {F}}_p$ can be generated, which are constant on the sets of a partition of ${\mathbb {V}}_{2m}^{(p)}$ 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 ${\mathbb {V}}_{2m}^{(p)}$ to B, where B can be any abelian group of order $p^k$, $k\le m$. As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of ${\mathbb {V}}_{2m}^{(2)}$, with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for ${\mathbb {V}}_{2m}^{(p)}$, 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 ${\mathbb {V}}_{2m}^{(p)}$ into a cyclic group ${\mathbb {Z}}_{p^k}$. With these results, we obtain the first constructions of bent functions from ${\mathbb {V}}_{2m}^{(p)}$ into ${\mathbb {Z}}_{p^k}$, p odd, which provably do not come from (partial) spreads.