432 research outputs found
Advanced Probabilistic Couplings for Differential Privacy
Differential privacy is a promising formal approach to data privacy, which
provides a quantitative bound on the privacy cost of an algorithm that operates
on sensitive information. Several tools have been developed for the formal
verification of differentially private algorithms, including program logics and
type systems. However, these tools do not capture fundamental techniques that
have emerged in recent years, and cannot be used for reasoning about
cutting-edge differentially private algorithms. Existing techniques fail to
handle three broad classes of algorithms: 1) algorithms where privacy depends
accuracy guarantees, 2) algorithms that are analyzed with the advanced
composition theorem, which shows slower growth in the privacy cost, 3)
algorithms that interactively accept adaptive inputs.
We address these limitations with a new formalism extending apRHL, a
relational program logic that has been used for proving differential privacy of
non-interactive algorithms, and incorporating aHL, a (non-relational) program
logic for accuracy properties. We illustrate our approach through a single
running example, which exemplifies the three classes of algorithms and explores
new variants of the Sparse Vector technique, a well-studied algorithm from the
privacy literature. We implement our logic in EasyCrypt, and formally verify
privacy. We also introduce a novel coupling technique called \emph{optimal
subset coupling} that may be of independent interest
Proving uniformity and independence by self-composition and coupling
Proof by coupling is a classical proof technique for establishing
probabilistic properties of two probabilistic processes, like stochastic
dominance and rapid mixing of Markov chains. More recently, couplings have been
investigated as a useful abstraction for formal reasoning about relational
properties of probabilistic programs, in particular for modeling
reduction-based cryptographic proofs and for verifying differential privacy. In
this paper, we demonstrate that probabilistic couplings can be used for
verifying non-relational probabilistic properties. Specifically, we show that
the program logic pRHL---whose proofs are formal versions of proofs by
coupling---can be used for formalizing uniformity and probabilistic
independence. We formally verify our main examples using the EasyCrypt proof
assistant
Probabilistic Couplings For Probabilistic Reasoning
This thesis explores proofs by coupling from the perspective of formal verification. Long employed in probability theory and theoretical computer science, these proofs construct couplings between the output distributions of two probabilistic processes. Couplings can imply various probabilistic relational properties, guarantees that compare two runs of a probabilistic computation.
To give a formal account of this clean proof technique, we first show that proofs in the program logic pRHL (probabilistic Relational Hoare Logic) describe couplings. We formalize couplings that establish various probabilistic properties, including distribution equivalence, convergence, and stochastic domination. Then we deepen the connection between couplings and pRHL by giving a proofs-as-programs interpretation: a coupling proof encodes a probabilistic product program, whose properties imply relational properties of the original two programs. We design the logic xpRHL (product pRHL) to build the product program, with extensions to model more advanced constructions including shift coupling and path coupling.
We then develop an approximate version of probabilistic coupling, based on approximate liftings. It is known that the existence of an approximate lifting implies differential privacy, a relational notion of statistical privacy. We propose a corresponding proof technique---proof by approximate coupling---inspired by the logic apRHL, a version of pRHL for building approximate liftings. Drawing on ideas from existing privacy proofs, we extend apRHL with novel proof rules for constructing new approximate couplings. We give approximate coupling proofs of privacy for the Report-noisy-max and Sparse Vector mechanisms, well-known algorithms from the privacy literature with notoriously subtle privacy proofs, and produce the first formalized proof of privacy for these algorithms in apRHL.
Finally, we enrich the theory of approximate couplings with several more sophisticated constructions: a principle for showing accuracy-dependent privacy, a generalization of the advanced composition theorem from differential privacy, and an optimal approximate coupling relating two subsets of samples. We also show equivalences between approximate couplings and other existing definitions. These ingredients support the first formalized proof of privacy for the Between Thresholds mechanism, an extension of the Sparse Vector mechanism
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