Paper 2003/204 Original: eprint.iacr.org/2003/204 ↗

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. 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 quadratic form over field with characteristic 2 to get an nm-variable Boolean function with beautiful spectrum 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.