The Lower Bounds on the Second Order Nonlinearity of Cubic Boolean Functions

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Even the lower bounds on the second order nonlinearity is known only for a few particular functions. We investigate the lower bounds on the second order nonlinearity of cubic Boolean functions Fu(x) = Tr( Pm l=1 μlx l), where ul ∈ F ∗ 2n , dl = 2l + 2l + 1, il and jl are positive integers, n > il > jl. Especially, for a class of Boolean functions Gu(x) = Tr( Pm l=1 μlx l), we deduce a tighter lower bound on the second order nonlinearity of the functions, where ul ∈ F ∗ 2n , dl = 2 ilγ + 2l + 1, il > jl and γ 6= 1 is a positive integer such that gcd(n, γ) = 1. The lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx 2i+2j+1), μ ∈ F ∗ 2n , i and j are positive integers such that i > j, have recently (2009) been obtained by Gode and Gangopadhvay. Our results have the advantages over those of Gode and Gangopadhvay as follows. We first extend the results from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Also, our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x).