Differential privacy is the standard method for privacy-preserving data analysis. The importance of having strong guarantees on the reliability of implementations of differentially private algorithms is widely recognized and has sparked fruitful research on formal methods. However, the design patterns and language features used in modern DP libraries as well as the classes of guarantees that the library designers wish to establish often fall outside of the scope of previous verification approaches. We introduce a program logic suitable for verifying differentially private implementations written in complex, general-purpose programming languages. Our logic has first-class support for reasoning about privacy budgets as a separation logic resource. The expressiveness of the logic and the target language allow our approach to handle common programming patterns used in the implementation of libraries for differential privacy, such as privacy filters and caching. While previous work has focused on developing guarantees for programs written in domain-specific languages or for privacy mechanisms in isolation, our logic can reason modularly about primitives, higher-order combinators, and interactive algorithms. We demonstrate the applicability of our approach by implementing a verified library of differential privacy mechanisms, including an online version of the Sparse Vector Technique, as well as a privacy filter inspired by the popular Python library OpenDP, which crucially relies on our ability to handle the combination of randomization, local state, and higher-order functions. We demonstrate that our specifications are general and reusable by instantiating them to verify clients of our library. All of our results have been foundationally verified in the Rocq Prover.

Properties such as provable security and correctness for randomized programs are naturally expressed relationally as approximate equivalences. As a result, a number of relational program logics have been developed to reason about such approximate equivalences of probabilistic programs. However, existing approximate relational logics are mostly restricted to first-order programs without general state. In this paper we develop Approxis, a higher-order approximate relational separation logic for reasoning about approximate equivalence of programs written in an expressive ML-like language with discrete probabilistic sampling, higher-order functions, and higher-order state. The Approxis logic recasts the concept of error credits in the relational setting to reason about relational approximation, which allows for expressive notions of modularity and composition, a range of new approximate relational rules, and an internalization of a standard limiting argument for showing exact probabilistic equivalences by approximation. We also use Approxis to develop a logical relation model that quantifies over error credits, which can be used to prove exact contextual equivalence . We demonstrate the flexibility of our approach on a range of examples, including the PRP/PRF switching lemma, IND$-CPA security of an encryption scheme, and a collection of rejection samplers. All of the results have been mechanized in the Coq proof assistant and the Iris separation logic framework.

We present Coneris, the first higher-order concurrent separation logic for reasoning about error probability bounds of higher-order concurrent probabilistic programs with higher-order state. To support modular reasoning about concurrent (non-probabilistic) program modules, state-of-the-art program logics internalize the classic notion of linearizability within the logic through the concept of logical atomicity . In Coneris, we extend this idea to probabilistic concurrent program modules by capturing a novel notion of randomized logical atomicity within the logic. To do so, Coneris utilizes presampling tapes and a novel probabilistic update modality to describe how state is changed probabilistically at linearization points. We demonstrate this approach by means of smaller synthetic examples and larger case studies. All of the presented results, including the meta-theory, have been mechanized in the Rocq prover and the Iris separation logic framework.

The field of high-assurance cryptography is quickly maturing, yet a unified foundational framework for end-to-end formal verification of efficient cryptographic implementations is still missing. To address this gap, we use the Coq proof assistant to formally connect three existing tools: (1) the Hacspec emergent cryptographic specification language; (2) the Jasmin language for efficient, high-assurance cryptographic implementations; and (3) the SSProve foundational verification framework for modular cryptographic proofs. We first connect Hacspec with SSProve by devising a new translation from Hacspec specifications to imperative SSProve code. We validate this translation by considering a second, more standard translation from Hacspec to purely functional Coq code and generate a proof of the equivalence between the code produced by the two translations. We further define a translation from Jasmin to SSProve, which allows us to formally reason in SSProve about efficient cryptographic implementations in Jasmin. We prove this translation correct in Coq with respect to Jasmin's operational semantics. Finally, we demonstrate the usefulness of our approach by giving a foundational end-to-end Coq proof of an efficient AES implementation. For this case study, we start from an existing Jasmin implementation of AES that makes use of hardware acceleration and prove that it conforms to a specification of the AES standard written in Hacspec. We use SSProve to formalize the security of the encryption scheme based on the Jasmin implementation of AES.

Probabilistic programs often trade accuracy for efficiency, and thus may, with a small probability, return an incorrect result. It is important to obtain precise bounds for the probability of these errors, but existing verification approaches have limitations that lead to error probability bounds that are excessively coarse, or only apply to first-order programs. In this paper we present Eris, a higher-order separation logic for proving error probability bounds for probabilistic programs written in an expressive higher-order language. Our key novelty is the introduction of error credits , a separation logic resource that tracks an upper bound on the probability that a program returns an erroneous result. By representing error bounds as a resource, we recover the benefits of separation logic, including compositionality, modularity, and dependency between errors and program terms, allowing for more precise specifications. Moreover, we enable novel reasoning principles such as expectation-preserving error composition, amortized error reasoning, and error induction. We illustrate the advantages of our approach by proving amortized error bounds on a range of examples, including collision probabilities in hash functions, which allow us to write more modular specifications for data structures that use them as clients. We also use our logic to prove correctness and almost-sure termination of rejection sampling algorithms. All of our results have been mechanized in the Coq proof assistant using the Iris separation logic framework and the Coquelicot real analysis library.

Almost-sure termination is an important correctness property for probabilistic programs, and a number of program logics have been developed for establishing it. However, these logics have mostly been developed for first-order programs written in languages with specific syntactic patterns for looping. In this paper, we consider almost-sure termination for higher-order probabilistic programs with general references. This combination of features allows for recursion and looping to be encoded through a variety of patterns. Therefore, rather than developing proof rules for reasoning about particular recursion patterns, we instead propose an approach based on proving refinement between a higher-order program and a simpler probabilistic model, in such a way that the refinement preserves termination behavior. By proving a refinement, almost-sure termination behavior of the program can then be established by analyzing the simpler model. We present this approach in the form of Caliper , a higher-order separation logic for proving terminationpreserving refinements. Caliper uses probabilistic couplings to carry out relational reasoning between a program and a model. To handle the range of recursion patterns found in higher-order programs, Caliper uses guarded recursion, in particular the principle of Löb induction. A technical novelty is that Caliper does not require the use of transfinite step indexing or other technical restrictions found in prior work on guarded recursion for termination-preservation refinement. We demonstrate the flexibility of this approach by proving almost-sure termination of several examples, including first-order loop constructs, a random list generator, treaps, and a sampler for Galton-Watson trees that uses higher-order store. All the results have been mechanized in the Coq proof assistant.

Probabilistic couplings are the foundation for many probabilistic relational program logics and arise when relating random sampling statements across two programs. In relational program logics, this manifests as dedicated coupling rules that, e.g ., say we may reason as if two sampling statements return the same value. However, this approach fundamentally requires aligning or “synchronizing” the sampling statements of the two programs which is not always possible. In this paper, we develop Clutch, a higher-order probabilistic relational separation logic that addresses this issue by supporting asynchronous probabilistic couplings. We use Clutch to develop a logical step-indexed logical relation to reason about contextual refinement and equivalence of higher-order programs written in a rich language with a probabilistic choice operator, higher-order local state, and impredicative polymorphism. Finally, we demonstrate our approach on a number of case studies. All the results that appear in the paper have been formalized in the Coq proof assistant using the Coquelicot library and the Iris separation logic framework.

We present Tachis, a higher-order separation logic to reason about the expected cost of probabilistic programs. Inspired by the uses of time credits for reasoning about the running time of deterministic programs, we introduce a novel notion of probabilistic cost credit. Probabilistic cost credits are a separation logic resource that can be used to pay for the cost of operations in programs, and that can be distributed across all possible branches of sampling instructions according to their weight, thus enabling us to reason about expected cost. The representation of cost credits as separation logic resources gives Tachis a great deal of flexibility and expressivity. In particular, it permits reasoning about amortized expected cost by storing excess credits as potential into data structures to pay for future operations. Tachis further supports a range of cost models, including running time and entropy usage. We showcase the versatility of this approach by applying our techniques to prove upper bounds on the expected cost of a variety of probabilistic algorithms and data structures, including randomized quicksort, hash tables, and meldable heaps. All of our results have been mechanized using Coq, Iris, and the Coquelicot real analysis library.

Abstract We give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin–Löf type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We prove several general meta-theorems about finitary type theories: weakening, admissibility of substitution and instantiation of metavariables, derivability of presuppositions, uniqueness of typing, and inversion principles. We then give a second formulation of finitary type theories in which there are no explicit contexts. Instead, free variables are explicitly annotated with their types. We provide translations between finitary type theories with and without contexts, thereby showing that they have the same expressive power. The context-free type theory is implemented in the nucleus of the Andromeda 2 proof assistant.

State-separating proofs (SSP) is a recent methodology for structuring game-based cryptographic proofs in a modular way, by using algebraic laws to exploit the modular structure of composed protocols. While promising, this methodology was previously not fully formalized and came with little tool support. We address this by introducing SSProve, the first general verification framework for machine-checked state-separating proofs. SSProve combines high-level modular proofs about composed protocols, as proposed in SSP, with a probabilistic relational program logic for formalizing the lower-level details, which together enable constructing machine-checked cryptographic proofs in the Coq proof assistant. Moreover, SSProve is itself fully formalized in Coq, including the algebraic laws of SSP, the soundness of the program logic, and the connection between these two verification styles. To illustrate SSProve, we use it to mechanize the simple security proofs of ElGamal and pseudo-random-function–based encryption. We also validate the SSProve approach by conducting two more substantial case studies: First, we mechanize an SSP security proof of the key encapsulation mechanism–data encryption mechanism (KEM-DEM) public key encryption scheme, which led to the discovery of an error in the original paper proof that has since been fixed. Second, we use SSProve to formally prove security of the sigma-protocol zero-knowledge construction, and we moreover construct a commitment scheme from a sigma-protocol to compare with a similar development in CryptHOL. We instantiate the security proof for sigma-protocols to give concrete security bounds for Schnorr’s sigma-protocol.

State-separating proofs (SSP) is a recent methodology for structuring game-based cryptographic proofs in a modular way. While very promising, this methodology was previously not fully formalized and came with little tool support. We address this by introducing SSProve, the first general verification framework for machine-checked state-separating proofs. SSProve combines high-level modular proofs about composed protocols, as proposed in SSP, with a probabilistic relational program logic for formalizing the lower-level details, which together enable constructing fully machine-checked crypto proofs in the Coq proof assistant. Moreover, SSProve is itself formalized in Coq, including the algebraic laws of SSP, the soundness of the program logic, and the connection between these two verification styles.