A Composition Construction of Bent-Like Boolean Functions from Quadratic Polynomials

Abstract: In this paper, we generalize the composition construction of Khoo et al. for highly nonlinear Boolean functions ([1]). We utilize general quadratic forms instead of the trace map in the construction. The construction composes an n-variable Boolean function and an m-variable F 2 quadratic form over to get an nm-variable Boolean function with beautiful spectrum n property and a doubled algebraic degree. Especially, the method is suitable to construct functions with 3-valued spectra (bent-like functions) or ones with better spectra (near-bent functions). Our proof technique is based on classification of quadratic forms over finite fields and enumeration of solutions of quadratic equations. We also prove the p-ary analogy of these results for odd prime p