Monomial Boolean functions with large high-order nonlinearities

Abstract

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher order nonlinearities of some monomial Boolean functions.We prove lower bounds on the second-order nonlinearities of functions trn(x7) and trn(x2r+3) where n=2r. Among all monomial Boolean functions, our bounds match the best second-order nonlinearity lower bounds by Carlet [IEEE Transactions on Information Theory 54(3), 2008] and Yan and Tang [Discrete Mathematics 343(5), 2020] for odd and even n, respectively. We prove a lower bound on the third-order nonlinearity for functions trn(x15), which is the best third-order nonlinearity lower bound. For any r, we prove that the r-th order nonlinearity of trn(x2r+1−1) is at least 2n−1−2(1−2−r)n+r2r−1−1−O(2n2). For r≪log2⁡n, this is the best lower bound among all explicit functions.