Measuring information leakage using generalized gain functions

This paper introduces g-leakage, a rich generalization of the min-entropy model of quantitative information flow. In g-leakage, the benefit that an adversary derives from a certain guess about a secret is specified using a gain function g. Gain functions allow a wide variety of operational scenarios to be modeled, including those where the adversary benefits from guessing a value close to the secret, guessing a part of the secret, guessing a property of the secret, or guessing the secret within some number of tries. We prove important properties of g-leakage, including bounds between min-capacity, g-capacity, and Shannon capacity. We also show a deep connection between a strong leakage ordering on two channels, C ₁ and C ₂, and the possibility of factoring C ₁ into C ₂C ₃, for some C ₃. Based on this connection, we propose a generalization of the Lattice of Information from deterministic to probabilistic channels.