On Higher-order Correlation Immunity and Higher Nonlinearity for a Class of Boolean Functions

Abstract

In this paper, we study the higher-order correlation immunity and the higher nonlinearity of Boolean functions, which is constructed by dividing the set of n-variable independent into two parts. With the propagation of a Boolean function, we construct a class of Boolean functions with 1-order algebraic immunity and higher-order correlation immunity, and reveal the relationship between the correlation immunity and 1-degree annihilator of Boolean functions. Meanwhile, with the lowest algebraic degree annihilator of Boolean functions, we also derive the invariance of the nonlinearity of Boolean functions with higher correlation immunity, and prove the existence of a class of a Boolean function with higher nonlinearity.