Comparing systems: max-case refinement orders and application to differential privacy

Quantitative Information Flow (QIF) and Differential Privacy (DP) are both concerned with the protection of sensitive information, but they are rather different approaches. In particular, QIF considers the expected probability of a successful attack, while DP (in both its standard and local versions) is a max-case measure, in the sense that it is compromised by the existence of a possible attack, regardless of its probability. Comparing systems is a fundamental task in these areas: one wishes to guarantee that replacing a system A by a system B is a safe operation, that is the privacy of B is no-worse than that of A. In QIF, a refinement order provides strong such guarantees, while in DP mechanisms are typically compared (w.r.t. privacy) based on the ϵ privacy parameter that they provide. In this paper we explore a variety of refinement orders, inspired by the one of QIF, providing precise guarantees for max-case leakage. We study simple structural ways of characterising them, the relation between them, efficient methods for verifying them and their lattice properties. Moreover, we apply these orders in the task of comparing DP mechanisms, raising the question of whether the order based on ϵ provides strong privacy guarantees. We show that, while it is often the case for mechanisms of the same 'family' (geometric, randomised response, etc.), it rarely holds across different families.