Generalized semifield spreads

Abstract

A (normal) bent partition of an n-dimensional vector space ${\mathbb {V}}_n^{(p)}$ over the prime field ${\mathbb {F}}_p$, is a partition of ${\mathbb {V}}_n^{(p)}$ into an n/2-dimensional subspace U, and subsets $A_1,\ldots , A_K$, such that every function $f:{\mathbb {V}}_n^{(p)}\rightarrow {\mathbb {F}}_p$ with the following property, is a bent function: The preimage set $f^{-1}(c) = \{x\in {\mathbb {V}}_n^{(p)}\,:\,f(x)=c\}$ contains exactly K/p of the sets $A_i$ for every $c\in {\mathbb {F}}_p$, and f is also constant on U. The classical examples are bent partitions from spreads or partial spreads, which have been known for a long time. Only recently (Meidl and Pirsic in Des Codes Cryptogr 89:75–89, 2021; Anbar and Meidl in Des Codes Cryptogr 90:1081–1101, 2022), it has been shown that (partial) spreads are not the only partitions with this remarkable property. Bent partitions have been presented, which generalize the Desarguesian spread, but provably do not come from any (partial) spread. In this article we show that also for some classes of semifields we can construct bent partitions, which similarly to finite fields and the Desarguesian spread, can be seen as a generalization of the semifield spread. Our results suggest that there are many partitions, which have similar properties as spreads.