On relationships among avalanche, nonlinearity, and correlation immunity

Abstract

We establish, for the first time, an explicit and simple lower bound on the nonlinearity Nf of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, Nf ≥ 2n−1 − 2n−1(Formula Presented)p. We also show that the lower bound is tight, and identify all the functions whose nonlinearity attains the lower bound. As a further contribution of this paper, we prove that except for very few cases, the sum of the degree of avalanche and the order of correlation immunity of a Boolean function of n variables is atmost n−2. These new results further highlight the significance of the fact that while avalanche property is in harmony with nonlinearity, it goes against correlation immunity.