Quadratic almost bent functions - Their partial characterization and design in the spectral domain

Abstract: Quadratic almost bent (AB) functions are characterized by the property that the duals of their component functions are bent functions. We prove that these duals are also quadratic and illustrate that these bent duals may give rise to vectorial bent functions (in certain cases having a maximal output dimension). A necessary and sufficient condition for ensuring bentness of the linear combinations of quadratic bent duals is provided. Moreover, we provide a rather detailed analysis related to the structure of quadratic AB functions in the spectral domain, more precisely with respect to their Walsh supports, their intersection and restrictions of these bent duals to suitable subspaces. In particular, we completely determine the intersection of Walsh supports of the coordinate (semi-bent) functions for Gold AB mappings. We also provide the design of quadratic AB functions in the spectral domain by identifying (using computer simulations) suitable sets of bent dual functions. For instance, when n=7, this approach provides several AB functions which are not CCZ-equivalent to Gold functions.