Further properties of several classes of Boolean functions with optimum algebraic immunity

Abstract

Based on a method proposed by the first author, several classes of balanced Boolean functions with optimum algebraic immunity are constructed, and they have nonlinearities significantly larger than the previously best known nonlinearity of functions with optimal algebraic immunity. By choosing suitable parameters, the constructed n-variable functions have nonlinearity \({2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}\Big/(n-2)}\) for even \({n\geq 8\,{\rm and}\,2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)}\) for odd n, where Δ(n) is a function increasing rapidly with n. The algebraic degrees of some constructed functions are also discussed.