4,975 research outputs found
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
Coupled Relational Symbolic Execution for Differential Privacy
Differential privacy is a de facto standard in data privacy with applications
in the private and public sectors. Most of the techniques that achieve
differential privacy are based on a judicious use of randomness. However,
reasoning about randomized programs is difficult and error prone. For this
reason, several techniques have been recently proposed to support designer in
proving programs differentially private or in finding violations to it. In this
work we propose a technique based on symbolic execution for reasoning about
differential privacy. Symbolic execution is a classic technique used for
testing, counterexample generation and to prove absence of bugs. Here we use
symbolic execution to support these tasks specifically for differential
privacy. To achieve this goal, we leverage two ideas that have been already
proven useful in formal reasoning about differential privacy: relational
reasoning and probabilistic coupling. Our technique integrates these two ideas
and shows how such a combination can be used to both verify and find violations
to differential privacy
Divergences on Monads for Relational Program Logics
Several relational program logics have been introduced for integrating
reasoning about relational properties of programs and measurement of
quantitative difference between computational effects. Towards a general
framework for such logics, in this paper, we formalize quantitative difference
between computational effects as divergence on monad, then develop a relational
program logic acRL that supports generic computational effects and divergences
on them. To give a categorical semantics of acRL supporting divergences, we
give a method to obtain graded strong relational liftings from divergences on
monads. We derive two instantiations of acRL for the verification of 1) various
differential privacy of higher-order functional probabilistic programs and 2)
difference of distribution of costs between higher-order functional programs
with probabilistic choice and cost counting operations.Comment: Preprin
Coupled relational symbolic execution for differential privacy
Differential privacy is a de facto standard in data privacy with applications in the private and public sectors. Most of the techniques that achieve differential privacy are based on a judicious use of randomness. However, reasoning about randomized programs is difficult and error prone. For this reason, several techniques have been recently proposed to support designer in proving programs differentially private or in finding violations to it.
In this work we propose a technique based on symbolic execution for reasoning about differential privacy. Symbolic execution is a classic technique used for testing, counterexample generation and to prove absence of bugs. Here we use symbolic execution to support these tasks specifically for differential privacy. To achieve this goal, we design a relational symbolic execution technique which supports reasoning about probabilistic coupling, a formal notion that has been shown useful to structure proofs of differential privacy. We show how our technique can be used to both verify and find violations to differential privacy.https://link.springer.com/content/pdf/10.1007%2F978-3-030-72019-3_8.pd
A program logic for union bounds
International audienceWe propose a probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory. While the union bound is simple, it is an extremely common tool for analyzing randomized algorithms. In formal verification terms, the union bound allows flexible and compos-itional reasoning over possible ways an algorithm may go wrong. It also enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in our logic. Notably, assertions in our logic are non-probabilistic, even though we can conclude probabilistic facts from the judgments. Our logic can also prove accuracy properties for interactive programs, where the program must produce intermediate outputs as soon as pieces of the input arrive, rather than accessing the entire input at once. This setting also enables adaptivity, where later inputs may depend on earlier intermediate outputs. We show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive. 1998 ACM Subject Classification D.2.4 Software/Program Verification 1 Introduction Probabilistic computations arise naturally in many areas of computer science. For instance, they are widely used in cryptography, privacy, and security for achieving goals that lie beyond the reach of deterministic programs. However, the correctness of probabilistic programs can be quite subtle, often relying on complex reasoning about probabilistic events. Accordingly, probabilistic computations present an attractive target for formal verification. A long line of research, spanning more than four decades, has focused on expressive formalisms for reasoning about general probabilistic properties both for purely probabilistic programs and for programs that combine probabilistic and non-deterministic choice (see, e.g., [29, 34, 35]). More recent research investigates specialized formalisms that work with more restricted assertions and proof techniques, aiming to simplify formal verification. As perhaps the purest examples of this approach, some program logics prove probabilistic properties by working purely with non-probabilistic assertions; we call such systems lightweight logics. Examples include probabilistic relational Hoare logic [3] for proving the reductionist security of cryptographic constructions, and the related approximate probabilistic relational Hoare logic [4] for reasoning about differential privacy. These logics rely on the powerful abstraction of probabilistic couplings to derive probabilistic facts from non-probabilistic assertions [7]
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
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