Bent partitions

• Published in: Designs, codes, and cryptography codes, and cryptography, 2022, Vol.90 (4), p.1081-1101
• Main Authors: Anbar, Nurdagül, Meidl, Wilfried
• Format: Article
• Language: English
• Subjects: Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Group theory; Subspaces
• ISSN: 0925-1022; 1573-7586
• DOI: 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 2 m -dimensional vector space V 2 m ( p ) over F p into F p can be generated, which are constant on the sets of a partition of V 2 m ( 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 V 2 m ( p ) 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 m ( 2 ) , with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for V 2 m ( 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 V 2 m ( p ) into a cyclic group Z p k . With these results, we obtain the first constructions of bent functions from V 2 m ( p ) into Z p k , p odd, which provably do not come from (partial) spreads.