Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices

TL;DR

This paper broadens the understanding of cocyclic Butson matrices and their connections to generalized bent functions, enabling new constructions of Boolean functions with specific cryptographic properties.

Contribution

It establishes new equivalences between cocyclic Butson matrices and generalized partially bent functions, extending prior group-invariant results.

Findings

Broader network of equivalences between cocyclic matrices and bent functions

Construction methods for Boolean functions with generalized partial bent properties

Applications to cases where no traditional bent functions exist

Abstract

In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications; for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist.