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, we tighten the lower bound of the second-order nonlinearity of a class of Boolean function over finite field F_2ⁿ, f_λ(x) = Tr(λx^d), where λ ∈ F^*_2r , d = 2^2r + 2^r + 1 and n = 7r. This bound is much better than the lower bound of Iwata-Kurosawa.