On second order nonlinearities of cubic monomial Boolean functions.

Abstract

We study cubic monomial Boolean functions of the form T rn 1 (µx2i +2j+1) where µ ∈ F2n. We prove that the functions of this form do not have any affine derivative if n ̸ = i + j or n ̸ = 2i − j. Lower bounds on the second order nonlinearities of these functions are derived.