Min-Entropy Leakage of Channels in Cascade

Theories of quantitative information flow offer an attractive framework for analyzing confidentiality in practical systems, which often cannot avoid "small" leaks of confidential information. Recently there has been growing interest in the theory of min-entropy leakage, which measures uncertainty based on a random variable's vulnerability to being guessed in one try by an adversary. Here we contribute to this theory by studying the min-entropy leakage of systems formed by cascading two channels together, using the output of the first channel as the input to the second channel. After considering the semantics of cascading carefully and exposing some technical subtleties, we prove that the min-entropy leakage of a cascade of two channels cannot exceed the leakage of the first channel; this result is a min-entropy analogue of the classic data-processing inequality. We show however that a comparable bound does not hold for the second channel. We then consider the min-capacity, or maximum leakage over all a priori distributions, showing that the min-capacity of a cascade of two channels cannot exceed the min-capacity of either channel.