Formal Verification of Differential Privacy for Interactive Systems

Differential privacy is a promising approach to privacy preserving data analysis with a well-developed theory for functions. Despite recent work on implementing systems that aim to provide differential privacy, the problem of formally verifying that these systems have differential privacy has not been adequately addressed. This paper presents the first results towards automated verification of source code for differentially private interactive systems. We develop a formal probabilistic automaton model of differential privacy for systems by adapting prior work on differential privacy for functions. The main technical result of the paper is a sound proof technique based on a form of probabilistic bisimulation relation for proving that a system modeled as a probabilistic automaton satisfies differential privacy. The novelty lies in the way we track quantitative privacy leakage bounds using a relation family instead of a single relation. We illustrate the proof technique on a representative automaton motivated by PINQ, an implemented system that is intended to provide differential privacy. To make our proof technique easier to apply to realistic systems, we prove a form of refinement theorem and apply it to show that a refinement of the abstract PINQ automaton also satisfies our differential privacy definition. Finally, we begin the process of automating our proof technique by providing an algorithm for mechanically checking a restricted class of relations from the proof technique.