Differentially Private Algorithms for Statistical Verification of Cyber-Physical Systems

Statistical model checking is a class of sequential algorithms that can verify specifications of interest on an ensemble of cyber-physical systems (e.g., whether 99% of cars from a batch meet a requirement on their functionality). These algorithms infer the probability that given specifications are satisfied by the systems with provable statistical guarantees by drawing sufficient numbers of independent and identically distributed samples. During the process of statistical model checking, the values of the samples (e.g., a user's car trajectory) may be inferred by intruders, causing privacy concerns in consumer-level applications (e.g., automobiles and medical devices). This paper addresses the privacy of statistical model checking algorithms from the point of view of differential privacy. These algorithms are sequential, drawing samples until a condition on their values is met. We show that revealing the number of samples drawn can violate privacy. We also show that the standard exponential mechanism that randomizes the output of an algorithm to achieve differential privacy fails to do so in the context of sequential algorithms. Instead, we relax the conservative requirement in differential privacy that the sensitivity of the output of the algorithm should be bounded to any perturbation for any data set. We propose a new notion of differential privacy which we call expected differential privacy (EDP). Then, we propose a novel expected sensitivity analysis for the sequential algorithm and propose a corresponding exponential mechanism that randomizes the termination time to achieve the EDP. We apply the proposed exponential mechanism to statistical model checking algorithms to preserve the privacy of the samples they draw. The utility of the proposed algorithm is demonstrated in a case study.