On Relationships among Avalanche, Nonlinearity, and Correlation Immunity

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 2 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 at most 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. Key Words: Avalanche Criterion, Boolean Functions, Correlation Immunity, Nonlinearity, Propagation Criterion.

• Type: Conference
• Year: 2000
• Where: ASIACRYPT
• Authors: Yuliang Zheng, Xian-Mo Zhang
• Keywords: ASIACRYPT 2000, Boolean Functions, Correlation Immunity, Cryptology, Lower Bound