Paper 2021/550

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

Amar Bapić, Samir Hodžić, and Enes Pasalic

Abstract

Quadratic AB (almost bent) 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). It is then natural to investigate when the linear combinations of quadratic bent duals again yield quadratic bent functions. A necessary and sufficient condition for ensuring the bentness of these linear combinations is provided, by introducing a useful transform that acts on the Walsh spectrum of dual functions. 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. It turns out that the AB property is quite complicated even in the quadratic case. However, using the established facts in this article, we could for the first time provide the design of quadratic AB functions in the spectral domain by identifying (using computer simulations) suitable sets of bent dual functions which give rise to possibly new quadratic AB functions in a generic manner. Using a simple non-exhaustive search for suitable sets of defining bent duals $f_1, \ldots,f_5$ on $\mathbb{F}_2^4$, we could easily identify 60 quadratic AB functions $F:\mathbb{F}_2^5 \rightarrow \mathbb{F}_2^5$. It turns out that all these functions are CCZ-equivalent to the Gold AB function but none of these functions is a permutation. On the other hand, when $n=7$, the same approach provides several AB functions which are \textbf{not} CCZ-equivalent to Gold functions.