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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-1-1/2p. 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.

On Relationships among Avalanche, Nonlinearity, and Correlation Immunity