Narrow Results

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions , where , d_l = 2^il + 2^jl + 1, m, i_l and j_l are positive integers, n > i_l > j_l. Furthermore, for a class of Boolean functions we deduce a tighter lower bound on the second order nonlinearity of the functions, where , d_l = 2^i_lγ + 2^j_lγ + 1, i_l > j_l and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1.

Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by f_μ(x) = Tr(μx^2i+2j+1), , i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions f_μ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

Inspired by a recent work of Mesnager, we present several infinite families of quadratic ternary bent, near-bent and 2-plateaued functions from some known quadratic ternary bent functions. Meanwhile, the distributions of the Walsh spectrum of two classes of 2-plateaued functions obtained in this paper are completely determined.

In this paper, we study the conjecture that $n$-variable ($n$ odd) rotation symmetric Boolean functions with degree $n-2$ have no non-zero linear structures. We show that if this class of RSBFs have non-zero linear structures, then the linear structures are invariant linear structures and the homogeneous component of degree $n-3$ in the function’s algebraic normal form has only two possibilities. Moreover, it is checked that the conjecture is true for $n=9,15,21$, and then a more explicit conjecture is proposed.