Bent partitions

Official URL: http://dx.doi.org/10.1007/s10623-022-01029-z

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 Fp into Fp 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 pk, 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 Zpk. With these results, we obtain the first constructions of bent functions from V(p)2m into Zpk, p odd, which provably do not come from (partial) spreads.