The lower bound of second-order nonlinearity of a class of Boolean functions

Abstract

The r-th nonlinearity of Boolean functions is an important cryptographic criterion associated with higher order linearity attacks on stream and block ciphers. In this paper, by investigating the low bound of the derivative of the Boolean function f, we tighten the lower bound of the second-order nonlinearity of a class of Boolean function over finite field F₂n, fγ(x) = Tr(Γx^d), where Γ ∈ F∗_2r.d = 2^2r+ 2^r+1 and n = 7r. This bound is is much better than the lower bound of Iwata-Kurosawa and can be used in stream ciphers as well as block ciphers. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.