Differential privacy is a notion of confidentiality that protects the privacy of individuals while allowing useful computations on their private data. Deriving differential privacy guarantees for real programs is a difficult and error-prone task that calls for principled approaches and tool support. Approaches based on linear types and static analysis have recently emerged; however, an increasing number of programs achieve privacy using techniques that cannot be analyzed by these approaches. Examples include programs that aim for weaker, approximate differential privacy guarantees, programs that use the Exponential mechanism, and randomized programs that achieve differential privacy without using any standard mechanism. Providing support for reasoning about the privacy of such programs has been an open problem.We report on CertiPriv, a machine-checked framework for reasoning about differential privacy built on top of the Coq proof assistant. The central component of CertiPriv is a quantitative extension of a probabilistic relational Hoare logic that enables one to derive differential privacy guarantees for programs from first principles. We demonstrate the expressiveness of CertiPriv using a number of examples whose formal analysis is out of the reach of previous techniques. In particular, we provide the first machine-checked proofs of correctness of the Laplacian and Exponential mechanisms and of the privacy of randomized and streaming algorithms from the recent literature.

Differential privacy is a notion of confidentiality that protects the privacy of individuals while allowing useful computations on their private data. Deriving differential privacy guarantees for real programs is a difficult and error-prone task that calls for principled approaches and tool support. Approaches based on linear types and static analysis have recently emerged; however, an increasing number of programs achieve privacy using techniques that cannot be analyzed by these approaches. Examples include programs that aim for weaker, approximate differential privacy guarantees, programs that use the Exponential mechanism, and randomized programs that achieve differential privacy without using any standard mechanism. Providing support for reasoning about the privacy of such programs has been an open problem.We report on CertiPriv, a machine-checked framework for reasoning about differential privacy built on top of the Coq proof assistant. The central component of CertiPriv is a quantitative extension of a probabilistic relational Hoare logic that enables one to derive differential privacy guarantees for programs from first principles. We demonstrate the expressiveness of CertiPriv using a number of examples whose formal analysis is out of the reach of previous techniques. In particular, we provide the first machine-checked proofs of correctness of the Laplacian and Exponential mechanisms and of the privacy of randomized and streaming algorithms from the recent literature.

This paper presents a wp-style calculus for obtaining bounds on the expected run-time of probabilistic programs. Its application includes determining the (possibly infinite) expected termination time of a probabilistic program and proving positive almost-sure terminationdoes a program terminate with probability one in finite expected time?We provide several proof rules for bounding the run-time of loops, and prove the soundness of the approach with respect to a simple operational model. We show that our approach is a conservative extension of Nielson's approach for reasoning about the run-time of deterministic programs. We analyze the expected run-time of some example programs including a one-dimensional random walk and the coupon collector problem.

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This paper presents a wp-style calculus for obtaining expectations on the outcomes of (mutually) recursive probabilistic programs. We provide several proof rules to derive one-and two-sided bounds for such expectations, and show the soundness of our wp-calculus with respect to a probabilistic pushdown automaton semantics. We also give a wp-style calculus for obtaining bounds on the expected runtime of recursive programs that can be used to determine the (possibly infinite) time until termination of such programs.