The constant-time programming discipline is commonly used to protect cryptographic libraries against side-channel attacks. However, it is hard to write constant-time code; moreover, compilers can introduce constant-time violations. Therefore, it is important to ensure that assembly code is constant-time. One approach is to show that source programs are constant-time, and that constant-timeness is preserved by compilation. In this paper, we explore the methodological soundness and scalability of the Decompile-then-Analyze approach, a less conventional alternative that has been suggested in the broader setting of static analysis. Informally, the Decompile-then-Analyze approach uses decompilers a front-end for static analysis tools. As a motivation for our study, we show that current decompilers eliminate CT vulnerabilities before CT analysis, leading to non-CT programs being accepted as constant-time. Independently, we provide constructed examples of non-CT, exploitable, programs that are accepted by two popular CT analysis tools; in both cases the culprit are program transformations that are used internally prior to CT analysis and eliminate CT violations. While our examples do not invalidate the general approach of these tools, they emphasize the need for studying the Decompile-then-Analyze approach. On the methodological side, we define the notion of CT transparency . Informally, a program transformation is CT transparent if does not eliminate nor introduce CT violations. We also provide general methods for proving that a transformation is CT transparent, and show that several transformations of interest are transparent. We also sketch an extension of CT transparency to speculative constant-time, which is used by cryptographic software as a protection against Spectre attacks. On the practical side, we build a CT-transparent version of the popular LLVM-based decompiler Ret Dec , and combine it with CT-LLVM, an existing CT verification tool for LLVM. We evaluate the resulting tool, called CT- Ret Dec on a benchmark set of real-world vulnerabilities in binaries, and show that the modifications had significant impact on how well CT- Ret Dec performs.

Constant-time (CT) verification tools are commonly used for detecting potential side-channel vulnerabilities in cryptographic libraries. Recently, a new class of tools, called speculative constant-time (SCT) tools, has also been used for detecting potential Spectre vulnerabilities. In many cases, these SCT tools have emerged as liftings of CT tools. However, these liftings are seldom defined precisely and are almost never analyzed formally. The goal of this paper is to address this gap, by developing formal foundations for these liftings, and to demonstrate that these foundations can yield practical benefits. Concretely, we introduce a program transformation, coined Speculation-Passing Style (SPS), for reducing SCT verification to CT verification. Essentially, the transformation instruments the program with a new input that corresponds to attacker-controlled predictions and modifies the program to follow them. This approach is sound and complete, in the sense that a program is SCT if and only if its SPS transform is CT. Thus, we can leverage existing CT verification tools to prove SCT; we illustrate this by combining SPS with three standard methodologies for CT verification, namely reducing it to noninterference, assertion safety, and dynamic taint analysis. We realize these combinations with three existing tools, EasyCrypt, Binsec/Rel , and CTGrind , and we evaluate them on Kocher’s benchmarks for Spectre-v1. Our results focus on Spectre-v1 in the standard CT leakage model; however, we also discuss applications of our method to other variants of Spectre and other leakage models.

Spectre attacks void the guarantees of constant-time cryptographic code by leaking secrets during speculative execution. Recent research shows that such code can be protected from Spectre-v1 attacks with minimal overhead, but leaves open the question of protecting against other Spectre variants. In this work, we design, validate, implement, and verify a new approach to protect cryptographic code against all known classes of Spectre attacks, in particular Spectre-RSB. Our approach combines a new value-dependent information-flow type system that ensures that no secrets leak even under speculative execution and a compiler transformation that enables it on the generated low-level code. We first prove the soundness of the type system and the correctness of the compiler transformation using the Coq proof assistant. We then implement our approach in the Jasmin framework for high-assurance cryptography and demonstrate that the overhead incurred by all Spectre protections is below 2% for most cryptographic primitives and reaches only about 5--7% for the more complex post-quantum key-encapsulationkmechanism Kyber.

Abstract Labelled Dirac notation is a formalism commonly used by physicists to represent many-body quantum systems and by computer scientists to assert properties of quantum programs. It is supported by a rich equational theory for proving equality between expressions in the language. These proofs are typically carried on pen-and-paper, and can be exceedingly long and error-prone. We introduce D-Hammer, the first tool to support automated equational proof for labelled Dirac notation. The salient features of D-Hammer include: an expressive, higher-order, dependently-typed language for labelled Dirac notation; an efficient normalization algorithm; and an optimized ++ implementation. We evaluate the implementation on representative examples from both plain and labelled Dirac notation. In the case of plain Dirac notation, we show that our implementation significantly outperforms DiracDec (Xu et al., POPL’25).

Jasmin is a programming language for high-speed and high-assurance cryptography. Correctness proofs of Jasmin programs are typically carried out deductively in EasyCrypt. This allows generality, modularity and composable reasoning, but does not scale well for low-level architecture-specific routines. CryptoLine offers a semi-automatic approach to formally verify algebraically-rich low-level cryptographic routines. CryptoLine proofs are self-contained: they are not integrated into higher-level formal verification developments. This paper shows how to soundly use CryptoLine to discharge subgoals in functional correctness proofs for complex Jasmin programs. We extend Jasmin with annotations and provide an automatic translation into a CryptoLine model, where most complex transformations are certified. We also formalize and implement the automatic extraction of the semantics of a CryptoLine proof to EasyCrypt. Our motivating use-case is the X-Wing hybrid KEM, for which we present the first formally verified implementation.

We introduce a quantitative relational Hoare logic for quantum programs. Assertions of the logic range over a new infinitary extension of positive semidefinite operators. We prove that our logic is sound, and complete for bounded postconditions and almost surely terminating programs. Our completeness result is based on a quantum version of the duality theorem from optimal transport. We also define a complete embedding into our logic of a relational Hoare logic with projective assertions.

We propose a hybrid formal verification approach that combines high-level deductive reasoning and circuit-based reasoning and apply it to highly optimized cryptographic assembly code. Our approach permits scaling up formal verification in two complementary directions: 1) it reduces the proof effort required for low-level functions where the computation logics are obfuscated by the intricate use of architecture-specific instructions and 2) it permits amortizing the effort of proving one implementation by using equivalence checking to propagate the guarantees to other implementations of the same computation using different optimizations or targeting different architectures. We demonstrate our approach via an extension to the EasyCrypt proof assistant and by revisiting formally verified implementations of ML-KEM in Jasmin. As a result, we obtain the first formally verified implementation of ML-KEM that offers performance comparable to the fastest non-verified implementation in x86-64 architectures.

We study expected runtimes for quantum programs. Inspired by recent work on probabilistic programs, we first define expected runtime as a generalisation of quantum weakest precondition . Then, we show that the expected runtime of a quantum program should be represented as the expectation of an observable (in physics). A method for computing the expected runtimes of quantum programs in finite-dimensional state spaces is developed. Several examples are provided as applications of this method, including computing the expected runtime of quantum Bernoulli Factory – a quantum algorithm for generating random numbers. In particular, using our new method, an open problem of computing the expected runtime of quantum random walks introduced by Ambainis et al. ( STOC 2001) is solved.

We introduce eRHL , a program logic for reasoning about relational expectation properties of pairs of probabilistic programs. eRHL is quantitative, i.e., its pre- and post-conditions take values in the extended non-negative reals. Thanks to its quantitative assertions, eRHL overcomes randomness alignment restrictions from prior logics, including pRHL , a popular relational program logic used to reason about security of cryptographic constructions, and apRHL , a variant of pRHL for differential privacy. As a result, eRHL is the first relational probabilistic program logic to be supported by non-trivial soundness and completeness results for all almost surely terminating programs. We show that eRHL is sound and complete with respect to program equivalence, statistical distance, and differential privacy. We also show that every pRHL judgment is valid iff it is provable in eRHL . We showcase the practical benefits of eRHL with examples that are beyond reach of pRHL and apRHL .

Compilers often weaken or even discard software-based countermeasures commonly used to protect programs against side-channel attacks; worse, they may also introduce vulnerabilities that attackers can exploit. The solution to this problemis to develop compilers that preserve such countermeasures. Prior work establishes that (a mildly modified version of) the CompCert and Jasmin formally verified compilers preserve constant-time, an information flow policy that ensures that programs are protected against timing side-channel attacks. However, nothing is known about preservation of speculative constant-time, a strengthening of the constant-time policythat ensures that programs are protected against Spectre-v1 attacks. We first show that preservation of speculative constant-time fails in practice by providing examples of secure programs whose compilation is not speculative constant-time using GCC (GCC -O0 and GCC -O1) and Jasmin. Then, we define a proof-of-concept compiler that distills some of the critical passes of the Jasmin compiler and use the C oq proof assistant to prove that it preserves speculative constant-time. Finally, we patch the Jasmin speculative constant-time type checker and demonstrate that all cryptographic implementations written in Jasmin can be fixed with minimal impact.

Dirac notation is widely used in quantum physics and quantum programming languages to define, compute and reason about quantum states. This paper considers Dirac notation from the perspective of automated reasoning. We prove two main results: first, the first-order theory of Dirac notation is decidable, by a reduction to the theory of real closed fields and Tarski’s theorem. Then, we prove that validity of equations can be decided efficiently, using term-rewriting techniques. We implement our equivalence checking algorithm in Mathematica, and showcase its efficiency across more than 100 examples from the literature.

How will future microarchitectures impact the security of existing cryptographic implementations? As we cannot keep reducing the size of transistors, chip vendors have started developing new microarchitectural optimizations to speed up computation. A recent study (Sanchez Vicarte et al., ISCA 2021) suggests that these optimizations might open the Pandora's box of microarchitectural attacks. However, there is little guidance on how to evaluate the security impact of future optimization proposals.

We propose, implement, and evaluate a hopping proof approach for proving expectation-based properties of probabilistic programs. Our approach combines EHL, a syntax-directed proof system for reducing proof goals of a program to proof goals of simpler programs, with a "hopping" proof rule for reducing proof goals of an original program to proof goal of a different program which is suitably related (by means of pRHL, a relational program logic for probabilistic program) to the original program. We prove that EHL is sound for a core language with procedure calls and adversarial computations, and complete for the adversary-free fragment of the language. We also provide an implementation of EHL into EasyCrypt, a proof assistant tailored for reasoning about relational properties of probabilistic programs. We provide a tight integration of EHL with other program logics supported by EasyCrypt, and in particular probabilistic Relational Hoare Logic (pRHL). Using this tight integration, we give mechanized proofs of expected complexity of in-place implementations of randomized quickselect and skip lists. We also sketch applications of our approach to cryptographic proofs and discuss the broader impact of EHL in the EasyCrypt proof assistant.

The design of online learning algorithms typically aims to optimise the incurred loss or cost , e.g., the number of classification mistakes made by the algorithm. The goal of this paper is to build a type-theoretic framework to prove that a certain algorithm achieves its stated bound on the cost. Online learning algorithms often rely on randomness, their loss functions are often defined as expectations, precise bounds are often non-polynomial (e.g., logarithmic) and proofs of optimality often rely on potentialbased arguments. Accordingly, we present pλ-amor, a type-theoretic graded modal framework for analysing (expected) costs of higher-order probabilistic programs with recursion. pλ-amor is an effect-based framework which uses graded modal types to represent potentials, cost and probability at the type level. It extends prior work ( λ-amor) on cost analysis for deterministic programs. We prove pλ-amor sound relative to a Kripke step-indexed model which relates potentials with probabilistic coupling. We use pλ-amor to prove cost bounds of several examples from the online machine learning literature. Finally, we describe an extension of pλ-amor with a graded comonad and describe the relationship between the different modalities.

The formal analysis of cryptographic protocols traditionally focuses on trace and equivalence properties, for which decision procedures in the symbolic (or Dolev-Yao, or DY) model are known. However, many relevant security properties are expressed as DY hyperproperties that involve quantifications over both execution paths and attacker computations (which are constrained by the attacker’s knowledge in the underlying model of computation). DY hyperproperties generalise hyperproperties, for which many decision procedures exist, to the setting of DY models. Unfortunately, the subtle interactions between both forms of quantifications have been an obstacle to lifting decision procedures from hyperproperties to DY hyperproperties. The central contribution of the paper is the first procedure for deciding DY hyperproperties, in the usual setting where the number of protocol sessions is bounded and where the equational theory modelling cryptography is subterm-convergent. We prove that our decision procedure can decide the validity of any hyperproperty in which quantifications over messages are guarded and quantifications over attacker computations are limited to expressing the attacker’s knowledge. We also establish the complexity of the decision problem for several important fragments of the hyperlogic. Further, we illustrate the techniques and scope of our contributions through examples of related hyperproperties.

Practical information-flow programming languages commonly allow controlled leakage via a declassify construct—programmers can use this construct to declare intentional leakage. For instance, cryptographic signatures and ciphertexts, which are computed from private keys, are viewed as secret by information-flow analyses. Cryptographic libraries can use declassify to make this data public, as it is no longer sensitive.In this paper, we study the interaction between speculative execution and declassification. We show that speculative execution leads to unintended leakage from declassification sites. Concretely, we present a PoC that recovers keys from AES implementations. Our PoC is an instance of a Spectre attack, and remains effective even when programs are compiled with speculative load hardening (SLH), a widespread compiler-based countermeasure against Spectre. We develop formal countermeasures against these attacks, including a significant improvement to SLH we term selective speculative load hardening (selSLH). These countermeasures soundly enforce relative non-interference (RNI): Informally, the speculative leakage of a protected program is limited to the existing sequential leakage of the original program. We implement our simplest countermeasure in the FaCT language and compiler—which is designed specifically for high-assurance cryptography—and we see performance overheads of at most 10%. Finally, although we do not directly implement selSLH, our preliminary evaluation suggests a significant reduction in performance cost for cryptographic functions as compared to traditional SLH.

The current gold standard of cryptographic software is to write efficient libraries with systematic protections against timing attacks. In order to meet this goal, cryptographic engineers increasingly use high-assurance cryptography tools. These tools guide programmers and provide rigorous guarantees that can be verified independently by library users. However, high-assurance tools reason about overly simple execution models that elide transient execution leakage. Thus, implementations validated by high-assurance cryptography tools remain potentially vulnerable to transient execution attacks such as Spectre or Meltdown. Moreover, proposed countermeasures are not used in practice due to performance overhead.We propose, analyze, implement and evaluate an approach for writing efficient cryptographic implementations that are protected against Spectre v1 attacks. Our approach ensures speculative constant-time, an information flow property which guarantees that programs are protected against Spectre v1. Speculative constant-time is enforced by means of a (value-dependent) information flow type system. The type system tracks security levels depending on whether execution is misspeculating. We implement our approach in the Jasmin framework for high-assurance cryptography, and use it for protecting all implementations of an experimental cryptographic library that includes highly optimized implementations of symmetric primitives, of elliptic-curve cryptography, and of Kyber, a lattice-based KEM recently selected by NIST for standardization. The performance impact of our protections is very low; for example, less than 1% for Kyber and essentially zero for X25519.

CoqQ is a framework for reasoning about quantum programs in the Coq proof assistant. Its main components are: a deeply embedded quantum programming language, in which classic quantum algorithms are easily expressed, and an expressive program logic for proving properties of programs. CoqQ is foundational: the program logic is formally proved sound with respect to a denotational semantics based on state-of-art mathematical libraries (MathComp and MathComp Analysis). CoqQ is also practical: assertions can use Dirac expressions, which eases concise specifications, and proofs can exploit local and parallel reasoning, which minimizes verification effort. We illustrate the applicability of CoqQ with many examples from the literature.

In this work, we enhance the EasyCrypt proof assistant to reason about the computational complexity of adversaries. The key technical tool is a Hoare logic for reasoning about computational complexity (execution time and oracle calls) of adversarial computations. Our Hoare logic is built on top of the module system used by EasyCrypt for modeling adversaries. We prove that our logic is sound w.r.t. the semantics of EasyCrypt programs—we also provide full semantics for the EasyCrypt module system, which was lacking previously. We showcase (for the first time in EasyCrypt and in other computer-aided cryptographic tools) how our approach can express precise relationships between the probability of adversarial success and their execution time. In particular, we can quantify existentially over adversaries in a complexity class and express general composition statements in simulation-based frameworks. Moreover, such statements can be composed to derive standard concrete security bounds for cryptographic constructions whose security is proved in a modular way. As a main benefit of our approach, we revisit security proofs of some well-known cryptographic constructions and present a new formalization of universal composability.

We propose automated methods for synthesising attacks against indifferentiability, a powerful simulation-based notion of security commonly used to reason about symmetric-key constructions. Our methods are inspired from symbolic cryptography which is popular to reason about, e.g., cryptographic protocols. For that, we introduce a core programming language for algebraic distinguishers and study the class of universal distinguishers, who win the indifferentiability game against every simulator; then, we show that the universality of algebraic distinguishers can be reduced to solving systems of algebraic, deducibility and static-equivalence constraints.

Timed cryptography refers to cryptographic primitives designed to meet their security goals only for a short (polynomial) amount of time. Popular examples include timed commitments and verifiable delay functions. Such primitives are commonly used to guarantee fairness in multiparty protocols ("either none or all parties obtain the output of the protocol'') without relying on any trusted party. Despite their recent surge in popularity, timed cryptographic protocols remain out of scope of current symbolic verification tools, which idealise cryptographic primitives as algebraic operations, and thus do not consider fine-grained notions of time.

Cryptographic constant-time (CT) is a popular programming discipline used by cryptographic libraries to protect themselves against timing attacks. The CT discipline aims to enforce that program execution does not leak secrets, where leakage is defined by a formal leakage model. In practice, different leakage models coexist, sometimes even within a single library, both to reflect different architectures and to accommodate different security-efficiency trade-offs.

We study expected runtimes for quantum programs. Inspired by recent work on probabilistic programs, we first define expected runtime as a generalisation of quantum weakest precondition. Then, we show that the expected runtime of a quantum program can be represented as the expectation of an observable (in physics). A method for computing the expected runtimes of quantum programs in finite-dimensional state spaces is developed. Several examples are provided as applications of this method, including computing the expected runtime of quantum Bernoulli Factory – a quantum algorithm for generating random numbers. In particular, using our new method, an open problem of computing the expected runtime of quantum random walks introduced by Ambainis et al. (STOC 2001) is solved.

Spectre vulnerabilities violate our fundamental assumptions about architectural abstractions, allowing attackers to steal sensitive data despite previously state-of-the-art countermeasures. To defend against Spectre, developers of verification tools and compiler-based mitigations are forced to reason about microarchitectural details such as speculative execution. In order to aid developers with these attacks in a principled way, the research community has sought formal foundations for speculative execution upon which to rebuild provable security guarantees.This paper systematizes the community’s current knowledge about software verification and mitigation for Spectre. We study state-of-the-art software defenses, both with and without associated formal models, and use a cohesive framework to compare the security properties each defense provides. We explore a wide variety of tradeoffs in the expressiveness of formal frameworks, the complexity of defense tools, and the resulting security guarantees. As a result of our analysis, we suggest practical choices for developers of analysis and mitigation tools, and we identify several open problems in this area to guide future work on grounded software defenses.

Timing attacks are among the most devastating side-channel attacks, allowing remote attackers to retrieve secret material, including cryptographic keys, with relative ease. In principle, “these attacks are not that hard to mitigate the basic intuition, captured by the constant-time criterion, is that control-flow and memory accesses should be independent from secrets. Furthermore, there is a broad range of tools for automatically checking adherence to this intuition. Yet, these attacks still plague popular cryptographic libraries twenty-five years after their discovery, reflecting a dangerous gap between academic research and cryptographic engineering. This gap can potentially undermine the emerging shift towards high-assurance, formally verified cryptographic libraries. However, the causes for this gap remain uninvestigated. To understand the causes of this gap, we conducted a survey with 44 developers of 27 prominent open-source cryptographic libraries. The goal of the survey was to analyze if and how the developers ensure that their code executes in constant time. Our main findings are that developers are aware of timing attacks and of their potentially dramatic consequences and yet often prioritize other issues over the perceived huge investment of time and resources currently needed to make their code resistant to timing attacks. Based on the survey, we identify several shortcomings in existing analysis tools for constant-time, and issue recommendations that can make writing constant-time libraries less difficult. Our recommendations can inform future development of analysis tools, security-aware compilers, and cryptographic libraries, not only for constant-timeness, but in the broader context of side-channel attacks, in particular for micro-architectural side-channel attacks, which are a younger topic and too recent as focus for this survey.

We study the nature of applicative bisimilarity in λ-calculi endowed with operators for sampling from contin- uous distributions. On the one hand, we show that bisimilarity, logical equivalence, and testing equivalence all coincide with contextual equivalence when real numbers can be manipulated through continuous functions only. The key ingredient towards this result is a notion of Feller-continuity for labelled Markov processes, which we believe of independent interest, giving rise a broad class of LMPs for which coinductive and logically inspired equivalences coincide. On the other hand, we show that if no constraint is put on the way real numbers are manipulated, characterizing contextual equivalence turns out to be hard, and most of the aforementioned notions of equivalence are even unsound.

We enhance refinement types with mechanisms to reason about relational properties of probabilistic computations. Our mechanisms, which are inspired from probabilistic couplings, are applicable to a rich set of probabilistic properties, including expected sensitivity, which ensures that the distance between outputs of two probabilistic computations can be controlled from the distance between their inputs. We implement our mechanisms in the type system of Liquid Haskell and we use them to formally verify Haskell implementations of two classic machine learning algorithms: Temporal Difference (TD) reinforcement learning and stochastic gradient descent (SGD). We formalize a fragment of our system for discrete distributions and we prove soundness with respect to a set-theoretical semantics.

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

Counterfactual explanations (CFE) are being widely used to explain algorithmic decisions, especially in consequential decision-making contexts (e.g., loan approval or pretrial bail). In this context, CFEs aim to provide individuals affected by an algorithmic decision with the most similar individual (i.e., nearest individual) with a different outcome. However, while an increasing number of works propose algorithms to compute CFEs, such approaches either lack in optimality of distance (i.e., they do not return the nearest individual) and perfect coverage (i.e., they do not provide a CFE for all individuals); or they do not scale to complex models such as neural networks. In this work, we provide a framework based on Mixed-Integer Programming (MIP) to compute nearest counterfactual explanations for the outcomes of neural networks, with both provable guarantees and runtimes comparable to gradient-based approaches. Our experiments on the Adult, COMPAS, and Credit datasets show that, in contrast with previous methods, our approach allows for efficiently computing diverse CFEs with both distance guarantees and perfect coverage.

EasyCrypt is a formal verification tool used extensively for formalizing concrete security proofs of cryptographic constructions. However, the EasyCrypt formal logics consider only classical at- tackers, which means that post-quantum security proofs cannot be formalized and machine-checked with this tool. In this paper we prove that a natural extension of the EasyCrypt core logics permits capturing a wide class of post-quantum cryptography proofs, settling a question raised by (Unruh, POPL 2019). Leveraging our positive result, we implement EasyPQC, an extension of EasyCrypt for post-quantum security proofs, and use EasyPQC to verify post- quantum security of three classic constructions: PRF-based MAC, Full Domain Hash and GPV08 identity-based encryption.

In this paper we enhance the EasyCrypt proof assistant to reason about computational complexity of adversaries. The key technical tool is a Hoare logic for reasoning about computational complexity (execution time and oracle calls) of adversarial computations. Our Hoare logic is built on top of the module system used by EasyCrypt for modeling adversaries. We prove that our logic is sound w.r.t. the semantics of EasyCrypt programs --- we also provide full semantics for the EasyCrypt module system, which was previously lacking.

Many security properties of interest are captured by instrumented semantics that model the functional behavior and the leakage of programs. For several important properties, including cryptographic constant-time (CCT), leakage models are sufficiently abstract that one can define instrumented semantics for high-level and low-level programs. One important goal is then to relate leakage of source programs and leakage of their compilation---this can be used, e.g., to prove preservation of CCT. To simplify this task, we put forward the idea of structured leakage. In contrast to the usual modeling of leakage as a sequence of observations, structured leakage is tightly coupled with the operational semantics of programs. This coupling greatly simplifies the definition of leakage transformers that map the leakage of source programs to leakage of their compilation and yields more precise statements about the preservation of security properties. We illustrate our methods on the Jasmin compiler and prove preservation results for two policies of interest: CCT and cost.

Observational non-interference (ONI) is a generic information-flow policy for side-channel leakage. Informally, a program is ONI-secure if observing program leakage during execution does not reveal any information about secrets. Formally, ONI is parametrized by a leakage functionl, and different instances of ONI can be recovered through different instantiations ofl. One popular instance of ONI is the cryptographic constant-time (CCT) policy, which is widely used in cryptographic libraries to protect against timing and cache attacks. Informally, a program is CCT-secure if it does not branch on secrets and does not perform secret-dependent memory accesses. Another instance of ONI is the constant-resource (CR) policy, a relaxation of the CCT policy which is used in Amazon's s2n implementation of TLS and in several other security applications. Informally, a program is CR-secure if its cost (modelled by a tick operator over an arbitrary semi-group) does not depend on secrets.In this paper, we consider the problem of preserving ONI by compilation. Prior work on the preservation of the CCT policy develops proof techniques for showing that main compiler optimisations preserve the CCT policy. However, these proof techniques critically rely on the fact that the semi-group used for modelling leakage satisfies the property:l1+l1'=l2+l2'⇒l1=l2∧l1'=l2'Unfortunately, this non-cancelling property fails for the CR policy, because its underlying semi-group is (\mathbbN, +) and it is currently not known how to extend existing techniques to policies that do not satisfy non-cancellation.We propose a methodology for proving the preservation of the CR policy during a program transformation. We present an implementation of some elementary compiler passes, and apply the methodology to prove the preservation of these passes. Our results have been mechanically verified using the Coq proof assistant.

We propose a model of the substructural logic of Bunched Implications (BI) that is suitable for reasoning about quantum states. In our model, the separating conjunction of BI describes separable quantum states. We develop a program logic where pre- and post-conditions are BI formulas describing quantum states—the program logic can be seen as a counterpart of separation logic for imperative quantum programs. We exercise the logic for proving the security of quantum one-time pad and secret sharing, and we show how the program logic can be used to discover a flaw in Google Cirq’s tutorial on the Variational Quantum Algorithm (VQA).

Computer-aided cryptography is an active area of research that develops and applies formal, machine-checkable approaches to the design, analysis, and implementation of cryptography. We present a cross-cutting systematization of the computer-aided cryptography literature, focusing on three main areas: (i) design-level security (both symbolic security and computational security), (ii) functional correctness and efficiency, and (iii) implementation-level security (with a focus on digital side-channel resistance). In each area, we first clarify the role of computer-aided cryptography—how it can help and what the caveats are—in addressing current challenges. We next present a taxonomy of state-of-the-art tools, comparing their accuracy, scope, trustworthiness, and usability. Then, we highlight their main achievements, trade-offs, and research challenges. After covering the three main areas, we present two case studies. First, we study efforts in combining tools focused on different areas to consolidate the guarantees they can provide. Second, we distill the lessons learned from the computer-aided cryptography community’s involvement in the TLS 1.3 standardization effort. Finally, we conclude with recommendations to paper authors, tool developers, and standardization bodies moving forward.

High-assurance cryptography leverages methods from program verification and cryptography engineering to deliver efficient cryptographic software with machine-checked proofs of memory safety, functional correctness, provable security, and absence of timing leaks. Traditionally, these guarantees are established under a sequential execution semantics. However, this semantics is not aligned with the behavior of modern processors that make use of speculative execution to improve performance. This mismatch, combined with the high-profile Spectre-style attacks that exploit speculative execution, naturally casts doubts on the robustness of high-assurance cryptography guarantees. In this paper, we dispel these doubts by showing that the benefits of high-assurance cryptography extend to speculative execution, costing only a modest performance overhead. We build atop the Jasmin verification framework an end-to-end approach for proving properties of cryptographic software under speculative execution, and validate our approach experimentally with efficient, functionally correct assembly implementations of ChaCha20 and Poly1305, which are secure against both traditional timing and speculative execution attacks.

Sensitivity properties describe how changes to the input of a program affect the output, typically by upper bounding the distance between the outputs of two runs by a monotone function of the distance between the corresponding inputs. When programs are probabilistic, the distance between outputs is a distance between distributions. The Kantorovich lifting provides a general way of defining a distance between distributions by lifting the distance of the underlying sample space; by choosing an appropriate distance on the base space, one can recover other usual probabilistic distances, such as the Total Variation distance. We develop a relational pre-expectation calculus to upper bound the Kantorovich distance between two executions of a probabilistic program. We illustrate our methods by proving algorithmic stability of a machine learning algorithm, convergence of a reinforcement learning algorithm, and fast mixing for card shuffling algorithms. We also consider some extensions: using our calculus to show convergence of Markov chains to the uniform distribution over states and an asynchronous extension to reason about pairs of program executions with different control flow.

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

We define a continuation-passing style (CPS) translation for a typed λ-calculus with probabilistic choice, unbounded recursion, and a tick operator — for modeling cost. The target language is a (non-probabilistic) λ-calculus, enriched with a type of extended positive reals and a fixpoint operator. We then show that applying the CPS transform of an expression M to the continuation λ v . 0 yields the expected cost of M . We also introduce a formal system for higher-order logic, called EHOL, prove it sound, and show it can derive tight upper bounds on the expected cost of classic examples, including Coupon Collector and Random Walk. Moreover, we relate our translation to Kaminski et al.’s ert-calculus, showing that the latter can be recovered by applying our CPS translation to (a generalization of) the classic embedding of imperative programs into λ-calculus. Finally, we prove that the CPS transform of an expression can also be used to compute pre-expectations and to reason about almost sure termination.

Differential privacy is a mathematical framework for developing statistical computations with provable guarantees of privacy and accuracy. In contrast to the privacy component of differential privacy, which has a clear mathematical and intuitive meaning, the accuracy component of differential privacy does not have a generally accepted definition; accuracy claims of differential privacy algorithms vary from algorithm to algorithm and are not instantiations of a general definition. We identify program discontinuity as a common theme in existing ad hoc definitions and introduce an alternative notion of accuracy parametrized by, what we call, — the of an input x w.r.t. a deterministic computation f and a distance d , is the minimal distance d ( x , y ) over all y such that f ( y )≠ f ( x ). We show that our notion of accuracy subsumes the definition used in theoretical computer science, and captures known accuracy claims for differential privacy algorithms. In fact, our general notion of accuracy helps us prove better claims in some cases. Next, we study the decidability of accuracy. We first show that accuracy is in general undecidable. Then, we define a non-trivial class of probabilistic computations for which accuracy is decidable (unconditionally, or assuming Schanuel’s conjecture). We implement our decision procedure and experimentally evaluate the effectiveness of our approach for generating proofs or counterexamples of accuracy for common algorithms from the literature.

Differential privacy is a de facto standard in data privacy, with applicationsin the public and private sectors. One way of explaining differential privacy,which is particularly appealing to statistician and social scientists, is bymeans of its statistical hypothesis testing interpretation. Informally, onecannot effectively test whether a specific individual has contributed her databy observing the output of a private mechanism—any test cannot have bothhigh significance and high power.In this paper, we identify some conditions under which a privacy definition given in terms of a statistical divergence satisfies a similar interpretation.These conditions are useful to analyze the distinguishing power of divergencesand we use them to study the hypothesis testing interpretation of somerelaxations of differential privacy based on Renyi divergence. Ouranalysis also results in an improved conversion rule between these definitionsand differential privacy.

Predictive models are being increasingly used to support consequential decision making at the individual level in contexts such as pretrial bail and loan approval. As a result, there is increasing social and legal pressure to provide explanations that help the affected individuals not only to understand why a prediction was output, but also how to act to obtain a desired outcome. To this end, several works have proposed optimization-based methods to generate nearest counterfactual explanations. However, these methods are often restricted to a particular subset of models (e.g., decision trees or linear models) and differentiable distance functions. In contrast, we build on standard theory and tools from formal verification and propose a novel algorithm that solves a sequence of satisfiability problems, where both the distance function (objective) and predictive model (constraints) are represented as logic formulae. As shown by our experiments on real-world data, our algorithm is: i) model-agnostic ({non-}linear, {non-}differentiable, {non-}convex); ii) data-type-agnostic (heterogeneous features); iii) distance-agnostic (l0, l1, l8, and combinations thereof); iv) able to generate plausible and diverse counterfactuals for any sample (i.e., 100% coverage); and v) at provably optimal distances.

Abstract Logical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation , and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed $$\lambda $$ λ -calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued first-order functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.

Differential privacy is a de facto standard for statistical computations over databases that contain private data. Its main and rather surprising strength is to guarantee individual privacy and yet allow for accurate statistical results. Thanks to its mathematical definition, differential privacy is also a natural target for formal analysis. A broad line of work develops and uses logical methods for proving privacy. A more recent and complementary line of work uses statistical methods for finding privacy violations. Although both lines of work are practically successful, they elide the fundamental question of decidability.

We study decidability problems for equivalence of probabilistic programs, for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication and conditionals and thus is sufficiently expressive to encode boolean and arithmetic circuits. We consider two variants of equivalence: the first one considers an interpretation over the finite field Fq, while the second one, which we call universal equivalence, verifies equivalence over all extensions Fqk of Fq. The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F2k for any k. While the first problem is obviously decidable, we establish its exact complexity which lies in the counting hierarchy. To show decidability, and a doubly exponential upper bound, of the universal variant we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. Finally we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy.

We develop a new approach for building cryptographic implementations. Our approach goes the last mile and delivers assembly code that is provably functionally correct, protected against side-channels, and as efficient as hand-written assembly. We illustrate our approach using ChaCha20-Poly1305, one of the two ciphersuites recommended in TLS 1.3, and deliver formally verified vectorized implementations which outperform the fastest non-verified code.We realize our approach by combining the Jasmin framework, which offers in a single language features of high-level and low-level programming, and the EasyCrypt proof assistant, which offers a versatile verification infrastructure that supports proofs of functional correctness and equivalence checking. Neither of these tools had been used for functional correctness before. Taken together, these infrastructures empower programmers to develop efficient and verified implementations by "game hopping", starting from reference implementations that are proved functionally correct against a specification, and gradually introducing program optimizations that are proved correct by equivalence checking.We also make several contributions of independent interest, including a new and extensible verified compiler for Jasmin, with a richer memory model and support for vectorized instructions, and a new embedding of Jasmin in EasyCrypt. © 2020 IEEE.

Timing side-channels are arguably one of the main sources of vulnerabilities in cryptographic implementations. One effective mitigation against timing side-channels is to write programs that do not perform secret-dependent branches and memory accesses. This mitigation, known as "cryptographic constant-time", is adopted by several popular cryptographic libraries. This paper focuses on compilation of cryptographic constant-time programs, and more specifically on the following question: is the code generated by a realistic compiler for a constant-time source program itself provably constant-time? Surprisingly, we answer the question positively for a mildly modified version of the CompCert compiler, a formally verified and moderately optimizing compiler for C. Concretely, we modify the CompCert compiler to eliminate sources of potential leakage. Then, we instrument the operational semantics of CompCert intermediate languages so as to be able to capture cryptographic constant-time. Finally, we prove that the modified CompCert compiler preserves constant-time. Our mechanization maximizes reuse of the CompCert correctness proof, through the use of new proof techniques for proving preservation of constant-time. These techniques achieve complementary trade-offs between generality and tractability of proof effort, and are of independent interest.

Probabilistic independence is a useful concept for describing the result of random sampling—a basic operation in all probabilistic languages—and for reasoning about groups of random variables. Nevertheless, existing verification methods handle independence poorly, if at all. We propose a probabili stic separation logic PSL, where separation models probabilistic independence. We first give a new, probabilistic model of the logic of bunched implications (BI). We then build a program logic based on these assertions, and prove soundness of the proof system. We demonstrate our logic by verifying information-theoretic security of cryptographic constructions for several well-known tasks, including private information retrieval, oblivious transfer, secure multi-party addition, and simple oblivious RAM. Our proofs reason purely in terms of high-level properties, like independence and uniformity.

Relational verification of quantum programs has many potential applications in quantum and post-quantum security and other domains. We propose a relational program logic for quantum programs. The interpretation of our logic is based on a quantum analogue of probabilistic couplings. We use our logic to verify non-trivial relational properties of quantum programs, including uniformity for samples generated by the quantum Bernoulli factory, reliability of quantum teleportation against noise (bit and phase flip), security of quantum one-time pad and equivalence of quantum walks.

We present a high-assurance and high-speed implementation of the SHA-3 hash function. Our implementation is written in the Jasmin programming language, and is formally verified for functional correctness, provable security and timing attack resistance in the EasyCrypt proof assistant. Our implementation is the first to achieve simultaneously the four desirable properties (efficiency, correctness, provable security, and side-channel protection) for a non-trivial cryptographic primitive. Concretely, our mechanized proofs show that: 1) the SHA-3 hash function is indifferentiable from a random oracle, and thus is resistant against collision, first and second preimage attacks; 2) the SHA-3 hash function is correctly implemented by a vectorized x86 implementation. Furthermore, the implementation is provably protected against timing attacks in an idealized model of timing leaks. The proofs include new EasyCrypt libraries of independent interest for programmable random oracles and modular indifferentiability proofs.

In this paper, we propose a constant-time implementation of the BLISS lattice-based signature scheme. BLISS is possibly the most efficient lattice-based signature scheme proposed so far, with a level of performance on par with widely used pre-quantum primitives like ECDSA. It is only one of the few postquantum signatures to have seen real-world deployment, as part of the strongSwan VPN software suite. The outstanding performance of the BLISS signature scheme stems in large part from its reliance on discrete Gaussian distributions, which allow for better parameters and security reductions. However, that advantage has also proved to be its Achilles' heel, as discrete Gaussians pose serious challenges in terms of secure implementations. Implementations of BLISS so far have included secret-dependent branches and memory accesses, both as part of the discrete Gaussian sampling and of the essential rejection sampling step in signature generation. These defects have led to multiple devastating timing attacks, and were a key reason why BLISS was not submitted to the NIST postquantum standardization effort. In fact, almost all of the actual candidates chose to stay away from Gaussians despite their efficiency advantage, due to the serious concerns surrounding implementation security. Moreover, naive countermeasures will often not cut it: we show that a reasonable-looking countermeasure suggested in previous work to protect the BLISS rejection sampling can again be defeated using novel timing attacks, in which the timing information is fed to phase retrieval machine learning algorithm in order to achieve a full key recovery. Fortunately, we also present careful implementation techniques that allow us to describe an implementation of BLISS with complete timing attack protection, achieving the same level of efficiency as the original unprotected code, without resorting on floating point arithmetic or platform-specific optimizations like AVX intrinsics. These techniques, including a new approach to the polynomial approximation of transcendental function, can also be applied to the masking of the BLISS signature scheme, and will hopefully make more efficient and secure implementations of lattice-based cryptography possible going forward.

Code-based game-playing is a popular methodology for proving security of cryptographic constructions and side-channel countermeasures. This methodology relies on treating cryptographic proofs as an instance of relational program verification (between probabilistic programs), and decomposing the latter into a series of elementary relational program verification steps. In this paper, we develop principled methods for proving such elementary steps for probabilistic programs that operate over finite fields and related algebraic structures. We focus on three essential properties: program equivalence, information flow, and uniformity. We give characterizations of these properties based on deducibility and other notions from symbolic cryptography. We use (sometimes improve) tools from symbolic cryptography to obtain decision procedures or sound proof methods for program equivalence, information flow, and uniformity. Finally, we evaluate our approach using examples drawn from provable security and from side-channel analysis - for the latter, we focus on the masking countermeasure against differential power analysis. A partial implementation of our approach is integrated in EasyCrypt, a proof assistant for provable security, and in MaskVerif, a fully automated prover for masked implementations.

We present a logical framework for the verification of relational properties in imperative programs. Our frame-work reduces verification of relational properties of imperative programs to a validity problem in trace logic, an expressive instance of first-order predicate logic. Trace logic draws its expressiveness from its syntax, which allows expressing properties over computation traces. Its axiomatization supports fine-grained reasoning about intermediate steps in program execution, notably loop iterations. We present an algorithm to encode the semantics of programs as well as their relational properties in trace logic, and then show how first-order theorem proving can be used to reason about the resulting trace logic formulas. Our work is implemented in the tool RAPID and evaluated with examples coming from the security field.

We develop new abstractions for reasoning about three relaxations of differential privacy: Rényi differential privacy, zero-concentrated differential privacy, and truncated concentrated differential privacy, which express bounds on statistical divergences between two output probability distributions. In order to reason about such properties compositionally, we introduce approximate span-lifting, a novel construction extending the approximate relational lifting approaches previously developed for standard differential privacy to a more general class of divergences, and also to continuous distributions. As an application, we develop a program logic based on approximate span-liftings capable of proving relaxations of differential privacy and other statistical divergence properties.

A fundamental result in differential privacy states that the privacy guarantees of a mechanism are preserved by any post-processing of its output. In this paper we investigate under what conditions stochastic post-processing can amplify the privacy of a mechanism. By interpreting post-processing as the application of a Markov operator, we first give a series of amplification results in terms of uniform mixing properties of the Markov process defined by said operator. Next we provide amplification bounds in terms of coupling arguments which can be applied in cases where uniform mixing is not available. Finally, we introduce a new family of mechanisms based on diffusion processes which are closed under post-processing, and analyze their privacy via a novel heat flow argument. On the applied side, we generalize the analysis of "privacy amplification by iteration" in Noisy SGD and show it admits an exponential improvement in the strongly convex case, and study a mechanism based on the Ornstein–Uhlenbeck diffusion process which contains the Gaussian mechanism with optimal post-processing on bounded inputs as a special case.

Real-world cryptographic code is often written in a subset of C intended to execute in constant-time, thereby avoiding timing side channel vulnerabilities. This C subset eschews structured programming as we know it: if-statements, looping constructs, and procedural abstractions can leak timing information when handling sensitive data. The resulting obfuscation has led to subtle bugs, even in widely-used high-profile libraries like OpenSSL.

Relational type systems have been designed for several applications including information flow, differential privacy, and cost analysis. In order to achieve the best results, these systems often use relational refinements and relational effects to maximally exploit the similarity in the structure of the two programs being compared. Relational type systems are appealing for relational properties because they deliver simpler and more precise verification than what could be derived from typing the two programs separately. However, relational type systems do not yet achieve the practical appeal of their non-relational counterpart, in part because of the lack of a general foundation for implementing them.

Abstract Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security, or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called relational higher-order logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavor of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic, and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule, and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.

Recent developments in formal verification have identified approximate liftings (also known as approximate couplings) as a clean, compositional abstraction for proving differential privacy. This construction can be defined in two styles. Earlier definitions require the existence of one or more witness distributions, while a recent definition by Sato uses universal quantification over all sets of samples. These notions have each have their own strengths: the universal version is more general than the existential ones, while existential liftings are known to satisfy more precise composition principles. We propose a novel, existential version of approximate lifting, called $\star$-lifting, and show that it is equivalent to Sato's construction for discrete probability measures. Our work unifies all known notions of approximate lifting, yielding cleaner properties, more general constructions, and more precise composition theorems for both styles of lifting, enabling richer proofs of differential privacy. We also clarify the relation between existing definitions of approximate lifting, and consider more general approximate liftings based on $f$-divergences.

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on foundational semantics issues. In this paper, we take a step further by developing a suite of logics, collectively named PPV for proving properties of programs written in an expressive probabilistic higher-order language with continuous sampling operations and primitives for conditioning distributions. Our logics mimic the comfortable reasoning style of informal proofs using carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show expressiveness by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) ⊤⊤-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs.

Symbolic methods have been used extensively for proving security of cryptographic protocols in the Dolev-Yao model, and more recently for proving security of cryptographic primitives and constructions in the computational model. However, existing methods for proving security of cryptographic constructions in the computational model often require significant expertise and interaction, or are fairly limited in scope and expressivity. This paper introduces a symbolic approach for proving security of cryptographic constructions based on the Learning With Errors assumption (Regev, STOC 2005). Such constructions are instances of lattice-based cryptography and are extremely important due to their potential role in post-quantum cryptography. Following (Barthe, Grégoire and Schmidt, CCS 2015), our approach combines a computational logic and deducibility problems---a standard tool for representing the adversary's knowledge, the Dolev-Yao model. The computational logic is used to capture (indistinguishability-based) security notions and drive the security proofs whereas deducibility problems are used as side-conditions to control that rules of the logic are applied correctly. We then use AutoLWE, an implementation of the logic, to deliver very short or even automatic proofs of several emblematic constructions, including CPA-PKE (Gentry et al., STOC 2008), (Hierarchical) Identity-Based Encryption (Agrawal et al. Eurocrypt 2010), Inner Product Encryption (Agrawal et al. Asiacrypt 2011), CCA-PKE (Micciancio et al., Eurocrypt 2012). The main technical novelty beyond AutoLWE is a set of (semi-)decision procedures for deducibility problems, using extensions of Gröbner basis computations for subalgebras in the (non-)commutative setting (instead of ideals in the commutative setting). Our procedures cover the theory of matrices, which is required for lattice-based assumption, as well as the theory of non-commutative rings, fields, and Diffie-Hellman exponentiation, in its standard, bilinear and multilinear forms. Additionally, AutoLWE supports oracle-relative assumptions, which are used specifically to apply (advanced forms of) the Leftover Hash Lemma, an information-theoretical tool widely used in lattice-based proofs.

We give a language-based security treatment of domain-specific languages and compilers for secure multi-party computation, a cryptographic paradigm that enables collaborative computation over encrypted data. Computations are specified in a core imperative language, as if they were intended to be executed by a trusted-third party, and formally verified against an information-flow policy modelling (an upper bound to) their leakage. This allows non-experts to assess the impact of performance-driven authorized disclosure of intermediate values. Specifications are then compiled to multi-party protocols. We formalize protocol security using (distributed) probabilistic information-flow and prove security-preserving compilation: protocols only leak what is allowed by the source policy. The proof exploits a natural but previously missing correspondence between simulation-based cryptographic proofs and (composable) probabilistic non-interference. Finally, we extend our framework to justify leakage cancelling, a domain-specific optimization that allows to first write an efficient specification that fails to meet the allowed leakage upper-bound, and then apply a probabilistic pre-processing that brings leakage to the acceptable range.

Software-based countermeasures provide effective mitigation against side-channel attacks, often with minimal efficiency and deployment overheads. Their effectiveness is often amenable to rigorous analysis: specifically, several popular countermeasures can be formalized as information flow policies, and correct implementation of the countermeasures can be verified with state-of-the-art analysis and verification techniques. However, in absence of further justification, the guarantees only hold for the language (source, target, or intermediate representation) on which the analysis is performed. We consider the problem of preserving side-channel counter-measures by compilation for cryptographic “constant-time”, a popular countermeasure against cache-based timing attacks. We present a general method, based on the notion of constant-time-simulation, for proving that a compilation pass preserves the constant-time countermeasure. Using the Coq proof assistant, we verify the correctness of our method and of several representative instantiations.

We extend the simply-typed guarded $$\lambda $$ -calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types. The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.

We present Ellora , a sound and relatively complete assertion-based program logic, and demonstrate its expressivity by verifying several classical examples of randomized algorithms using an implementation in the EasyCrypt proof assistant. Ellora features new proof rules for loops and adversarial code, and supports richer assertions than existing program logics. We also show that Ellora allows convenient reasoning about complex probabilistic concepts by developing a new program logic for probabilistic independence and distribution law, and then smoothly embedding it into Ellora . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We define Almost Sure Productivity (ASP), a probabilistic generalization of the productivity condition for coinductively defined structures. Intuitively, a probabilistic coinductive stream or tree is ASP if it produces infinitely many outputs with probability 1. Formally, we define almost sure productivity using a final coalgebra semantics of programs inspired from Kerstan and König. Then, we introduce a core language for probabilistic streams and trees, and provide two approaches to verify ASP: a sufficient syntactic criterion, and a reduction to model-checking pCTL* formulas on probabilistic pushdown automata. The reduction shows that ASP is decidable for our core language.

Differential privacy comes equipped with multiple analytical tools for the design of private data analyses. One important tool is the so-called "privacy amplification by subsampling" principle, which ensures that a differentially private mechanism run on a random subsample of a population provides higher privacy guarantees than when run on the entire population. Several instances of this principle have been studied for different random subsampling methods, each with an ad-hoc analysis. In this paper we present a general method that recovers and improves prior analyses, yields lower bounds and derives new instances of privacy amplification by subsampling. Our method leverages a characterization of differential privacy as a divergence which emerged in the program verification community. Furthermore, it introduces new tools, including advanced joint convexity and privacy profiles, which might be of independent interest.

Program sensitivity, also known as Lipschitz continuity, describes how small changes in a program’s input lead to bounded changes in the output. We propose an average notion of program sensitivity for probabilistic programs—expected sensitivity—that averages a distance function over a probabilistic coupling of two output distributions from two similar inputs. By varying the distance, expected sensitivity recovers useful notions of probabilistic function sensitivity, including stability of machine learning algorithms and convergence of Markov chains. Furthermore, expected sensitivity satisfies clean compositional properties and is amenable to formal verification. We develop a relational program logic called EpRHL for proving expected sensitivity properties. Our logic features two key ideas. First, relational pre-conditions and post-conditions are expressed using distances, a real-valued generalization of typical boolean-valued (relational) assertions. Second, judgments are interpreted in terms of expectation coupling, a novel, quantitative generalization of probabilistic couplings which supports compositional reasoning. We demonstrate our logic on examples beyond the reach of prior relational logics. Our main example formalizes uniform stability of the stochastic gradient method. Furthermore, we prove rapid mixing for a probabilistic model of population dynamics. We also extend our logic with a transitivity principle for expectation couplings to capture the path coupling proof technique by Bubley and Dyer, and formalize rapid mixing of the Glauber dynamics from statistical physics.

Formal frameworks for cost analysis of programs have been widely studied in the unary setting and, to a limited extent, in the relational setting. However, many of these frameworks focus only on the cost aspect, largely side-lining functional properties that are often a prerequisite for cost analysis, thus leaving many interesting programs out of their purview. In this paper, we show that elegant, simple, expressive proof systems combining cost analysis and functional properties can be built by combining already known ingredients: higher-order refinements and cost monads. Specifically, we derive two syntax-directed proof systems, U C and R C , for unary and relational cost analysis, by adding a cost monad to a (syntax-directed) logic of higher-order programs. We study the metatheory of the systems, show that several nontrivial examples can be verified in them, and prove that existing frameworks for cost analysis (RelCost and RAML) can be embedded in them.

Jasmin is a framework for developing high-speed and high-assurance cryptographic software. The framework is structured around the Jasmin programming language and its compiler. The language is designed for enhancing portability of programs and for simplifying verification tasks. The compiler is designed to achieve predictability and efficiency of the output code (currently limited to x64 platforms), and is formally verified in the Coq proof assistant. Using the supercop framework, we evaluate the Jasmin compiler on representative cryptographic routines and conclude that the code generated by the compiler is as efficient as fast, hand-crafted, implementations. Moreover, the framework includes highly automated tools for proving memory safety and constant-time security (for protecting against cache-based timing attacks). We also demonstrate the effectiveness of the verification tools on a large set of cryptographic routines.

We present a high-assurance software stack for secure function evaluation (SFE). Our stack consists of three components: i. a verified compiler (CircGen) that translates C programs into Boolean circuits; ii. a verified implementation of Yao's SFE protocol based on garbled circuits and oblivious transfer; and iii. transparent application integration and communications via FRESCO, an open-source framework for secure multiparty computation (MPC). CircGen is a general purpose tool that builds on CompCert, a verified optimizing compiler for C. It can be used in arbitrary Boolean circuit-based cryptography deployments. The security of our SFE protocol implementation is formally verified using EasyCrypt, a tool-assisted framework for building high-confidence cryptographic proofs, and it leverages a new formalization of garbled circuits based on the framework of Bellare, Hoang, and Rogaway (CCS 2012). We conduct a practical evaluation of our approach, and conclude that it is competitive with state-of-the-art (unverified) approaches. Our work provides concrete evidence of the feasibility of building efficient, verified, implementations of higher-level cryptographic systems. All our development is publicly available.

Attribute-based encryption (ABE) is a cryptographic primitive which supports fine-grained access control on encrypted data, making it an appealing building block for many applications. In this paper, we propose, implement, and evaluate fully automated methods for proving security of ABE in the Generic Bilinear Group Model (Boneh, Boyen, and Goh, 2005, Boyen, 2008), an idealized model which admits simpler and more efficient constructions, and can also be used to find attacks. Our method is applicable to Rational-Fraction Induced ABE, a large class of ABE that contains most of the schemes from the literature, and relies on a Master Theorem, which reduces security in the GGM to a (new) notion of symbolic security, which is amenable to automated verification using constraint-based techniques. We relate our notion of symbolic security for Rational-Fraction Induced ABE to prior notions for Pair Encodings. Finally, we present several applications, including automated proofs for new schemes.

Motivated by applications to security and high efficiency, we propose an automated methodology for validating on low-level intermediate representations the results of a source-level static analysis. Our methodology relies on two main ingredients: a relative-safety checker, an instance of a relational verifier which proves that a program is "safer" than another, and a transformation of programs into defensive form which verifies the analysis results at runtime. We prove the soundness of the methodology, and provide a formally verified instantiation based on the Verasco verified C static analyzer and the CompCert verified C compiler. We experiment with the effectiveness of our approach with client optimizations at RTL level, and static analyses for cache-based timing side-channels and memory usage at pre-assembly levels.

Despite numerous countermeasures proposed by practitioners and researchers, remote control-flow alteration of programs with memory-safety vulnerabilities continues to be a realistic threat. Guaranteeing that complex software is completely free of memory-safety vulnerabilities is extremely expensive. Probabilistic countermeasures that depend on random secret keys are interesting, because they are an inexpensive way to raise the bar for attackers who aim to exploit memory-safety vulnerabilities. Moreover, some countermeasures even support legacy systems. However, it is unclear how to quantify and compare the effectiveness of different probabilistic countermeasures or combinations of such countermeasures. In this paper we propose a methodology to rigorously derive security bounds for probabilistic countermeasures. We argue that by representing security notions in this setting as events in probabilistic games, similarly as done with cryptographic security definitions, concrete and asymptotic guarantees can be obtained against realistic attackers. These guarantees shed light on the effectiveness of single countermeasures and their composition and allow practitioners to more precisely gauge the risk of an attack.

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.

Couplings are a powerful mathematical tool for reasoning about pairs of probabilistic processes. Recent developments in formal verification identify a close connection between couplings and pRHL, a relational program logic motivated by applications to provable security, enabling formal construction of couplings from the probability theory literature. However, existing work using pRHL merely shows existence of a coupling and does not give a way to prove quantitative properties about the coupling, needed to reason about mixing and convergence of probabilistic processes. Furthermore, pRHL is inherently incomplete, and is not able to capture some advanced forms of couplings such as shift couplings. We address both problems as follows.

Establishing quantitative bounds on the execution cost of programs is essential in many areas of computer science such as complexity analysis, compiler optimizations, security and privacy. Techniques based on program analysis, type systems and abstract interpretation are well-studied, but methods for analyzing how the execution costs of two programs compare to each other have not received attention. Naively combining the worst and best case execution costs of the two programs does not work well in many cases because such analysis forgets the similarities between the programs or the inputs.

Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called Relational Higher Order Logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavour of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.

Differential power analysis (DPA) is a side-channel attack in which an adversary retrieves cryptographic material by measuring and analyzing the power consumption of the device on which the cryptographic algorithm under attack executes. An effective countermeasure against DPA is to mask secrets by probabilistically encoding them over a set of shares, and to run masked algorithms that compute on these encodings. Masked algorithms are often expected to provide, at least, a certain level of probing security. Leveraging the deep connections between probabilistic information flow and probing security, we develop a precise, scalable, and fully automated methodology to verify the probing security of masked algorithms, and generate them from unprotected descriptions of the algorithm. Our methodology relies on several contributions of independent interest, including a stronger notion of probing security that supports compositional reasoning, and a type system for enforcing an expressive class of probing policies. Finally, we validate our methodology on examples that go significantly beyond the state-of-the-art.

We present PrivInfer, an expressive framework for writing and verifying differentially private Bayesian machine learning algorithms. Programs in PrivInfer are written in a rich functional probabilistic programming language with constructs for performing Bayesian inference. Then, differential privacy of programs is established using a relational refinement type system, in which refinements on probability types are indexed by a metric on distributions. Our framework leverages recent developments in Bayesian inference, probabilistic programming languages, and in relational refinement types. We demonstrate the expressiveness of PrivInfer by verifying privacy for several examples of private Bayesian inference.

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 on 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.

Over the last decade, differential privacy has achieved widespread adoption within the privacy community. Moreover, it has attracted significant attention from the verification community, resulting in several successful tools for formally proving differential privacy. Although their technical approaches vary greatly, all existing tools rely on reasoning principles derived from the composition theorem of differential privacy. While this suffices to verify most common private algorithms, there are several important algorithms whose privacy analysis does not rely solely on the composition theorem. Their proofs are significantly more complex, and are currently beyond the reach of verification tools.

Recent work on structure‐preserving signatures (SPS) studies optimality of these schemes in terms of the number of group elements needed in the verification key and the signature, and the number of pairing‐product equations in the verification algorithm. While these measures are crucial for many applications, another important aspect to consider for performance is verification time, which for these schemes is dominated by pairings computation. Although prior work considers optimality in terms of number of pairing‐product equations, this measure does not capture the exact number of pairings needed in verification. To fill this gap, we study the minimal number of pairings needed in verification of SPS. First, we prove lower bounds for schemes in the Type~II setting secure under chosen message attacks in the generic group model. We show that three pairings are necessary and at most one of these pairings can be precomputed. Second, we build an automated tool to search for schemes matching our lower bounds. Using this tool, we find a new randomisable SPS in the Type~II setting that is optimal with respect to our lower bound on the number of pairings, and minimal in terms of group operations to be computed during verification.

Analyzing cryptographic constructions in the computational model, or simply verifying the correctness of security proofs, are complex and error-prone tasks. Although computer tools have significant potential to increase confidence in security proofs and to reduce the time for building these proofs, existing tools are either limited in scope, or can only be used by formal methods experts, and have a significant overhead. In effect, it has remained a challenge to design usable and intuitive tools for building and verifying cryptographic proofs, especially for more advanced fields such as pairing-based or lattice-based cryptography.

Probabilistic coupling is a powerful tool for analyzing pairs of probabilistic processes. Roughly, coupling two processes requires finding an appropriate witness process that models both processes in the same probability space. Couplings are powerful tools proving properties about the relation between two processes, include reasoning about convergence of distributions and stochastic dominance —a probabilistic version of a monotonicity property. While the mathematical definition of coupling looks rather complex and cumbersome to manipulate, we show that the relational program logic pRHL—the logic underlying the EasyCrypt cryptographic proof assistant—already internalizes a generalization of probabilistic coupling. With this insight, constructing couplings is no harder than constructing logical proofs. We demonstrate how to express and verify classic examples of couplings in pRHL, and we mechanically verify several couplings in EasyCrypt. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mechanism design is the study of algorithm design where the inputs to the algorithm are controlled by strategic agents, who must be incentivized to faithfully report them. Unlike typical programmatic properties, it is not sufficient for algorithms to merely satisfy the property, incentive properties are only useful if the strategic agents also believe this fact.

Cache-based attacks are a class of side-channel attacks that are particularly effective in virtualized or cloud-based environments, where they have been used to recover secret keys from cryptographic implementations. One common approach to thwart cache-based attacks is to use constant-time implementations, i.e., which do not branch on secrets and do not perform memory accesses that depend on secrets. However, there is no rigorous proof that constant-time implementations are protected against concurrent cache-attacks in virtualization platforms with shared cache; moreover, many prominent implementations are not constant-time. An alternative approach is to rely on system-level mechanisms. One recent such mechanism is stealth memory, which provisions a small amount of private cache for programs to carry potentially leaking computations securely. Stealth memory induces a weak form of constant-time, called S-constant-time, which encompasses some widely used cryptographic implementations. However, there is no rigorous analysis of stealth memory and S-constant-time, and no tool support for checking if applications are S-constant-time.

Fault attacks are attacks in which an adversary with physical access to a cryptographic device, say a smartcard, tampers with the execution of an algorithm to retrieve secret material. Since the seminal Bellcore attack on modular exponentiation, there has been extensive work to discover new fault attacks against cryptographic schemes and develop countermeasures against such attacks. Originally focused on high-level algorithmic descriptions, these efforts increasingly focus on concrete implementations. While lowering the abstraction level leads to new fault attacks, it also makes their discovery significantly more challenging. In order to face this trend, it is therefore desirable to develop principled, tool-supported approaches that allow a systematic analysis of the security of cryptographic implementations against fault attacks.

Many algorithms admit very efficient batch versions that compute simultaneously the output of the algorithms on a set of inputs. Batch algorithms are widely used in cryptography, especially in the setting of pairing-based computations, where they deliver significant speed-ups. Auto Batch is an automated tool that computes highly optimized batch verification algorithms for pairing-based signature schemes. Thanks to finely tuned heuristics, Auto Batch is able to rediscover efficient batch verifiers for several signature schemes of interest, and in some cases to output batch verifiers that outperform the best known verifiers from the literature. However, Auto Batch only provides weak guarantees (in the form of a LaTeX proof) of the correctness of the batch algorithms it outputs. In this paper, we verify the correctness and security of these algorithms using the Easy Crypt framework. To achieve this goal, we define a domain-specific language to describe verification algorithms based on pairings and provide an efficient algorithm for checking (approximate) observational equivalence between expressions of this language. By translating the output of Auto Batch to this language and applying our verification procedure, we obtain machine-checked correctness proofs of the batch verifiers. Moreover, we formalize notions of security for batch verifiers and we provide a generic proof in Easy Crypt that batch verifiers satisfy a security property called screening, provided they are correct and the original signature is unforgeable against chosen-message attacks. We apply our techniques to several well-known pairing-based signature schemes from the literature, and to Groth-Sahai zero-knowledge proofs.

Differential privacy is a rigorous, worst-case notion of privacy-preserving computation. Informally, a probabilistic program is differentially private if the participation of a single individual in the input database has a limited effect on the program's distribution on outputs. More technically, differential privacy is a quantitative 2-safety property that bounds the distance between the output distributions of a probabilistic program on adjacent inputs. Like many 2-safety properties, differential privacy lies outside the scope of traditional verification techniques. Existing approaches to enforce privacy are based on intricate, non-conventional type systems, or customized relational logics. These approaches are difficult to implement and often cumbersome to use. We present an alternative approach that verifies differential privacy by standard, non-relational reasoning on non-probabilistic programs. Our approach transforms a probabilistic program into a non-probabilistic program which simulates two executions of the original program. We prove that if the target program is correct with respect to a Hoare specification, then the original probabilistic program is differentially private. We provide a variety of examples from the differential privacy literature to demonstrate the utility of our approach. Finally, we compare our approach with existing verification techniques for privacy.

Relational program logics have been used for mechanizing formal proofs of various cryptographic constructions. With an eye towards scaling these successes towards end-to-end security proofs for implementations of distributed systems, we present RF*, a relational extension of F*, a general-purpose higher-order stateful programming language with a verification system based on refinement types. The distinguishing feature of F* is a relational Hoare logic for a higher-order, stateful, probabilistic language. Through careful language design, we adapt the F* typechecker to generate both classic and relational verification conditions, and to automatically discharge their proofs using an SMT solver. Thus, we are able to benefit from the existing features of F*, including its abstraction facilities for modular reasoning about program fragments. We evaluate RF* experimentally by programming a series of cryptographic constructions and protocols, and by verifying their security properties, ranging from information flow to unlinkability, integrity, and privacy. Moreover, we validate the design of RF* by formalizing in Coq a core probabilistic λ calculus and a relational refinement type system and proving the soundness of the latter against a denotational semantics of the probabilistic lambda λ calculus.

CompCert is a formally verified compiler that generates compact and efficient code for a large subset of the C language. However, CompCert foregoes using SSA, an intermediate representation employed by many compilers that enables writing simpler, faster optimizers. In fact, it has remained an open problem to verify formally an SSA-based compiler. We report on a formally verified, SSA-based middle-end for CompCert. In addition to providing a formally verified SSA-based middle-end, we address two problems raised by Leroy in [2009]: giving an intuitive formal semantics to SSA, and leveraging its global properties to reason locally about program optimizations.

We present a computer-aided framework for proving concrete security bounds for cryptographic machine code implementations. The front-end of the framework is an interactive verification tool that extends the EasyCrypt framework to reason about relational properties of C-like programs extended with idealised probabilistic operations in the style of code-based security proofs. The framework also incorporates an extension of the CompCert certified compiler to support trusted libraries providing complex arithmetic calculations or instantiating idealized components such as sampling operations. This certified compiler allows us to carry to executable code the security guarantees established at the high-level, and is also instrumented to detect when compilation may interfere with side-channel countermeasures deployed in source code.

Computer-aided verification provides effective means of analyzing the security of cryptographic primitives. However, it has remained a challenge to achieve fully automated analyses yielding guarantees that hold against computational (rather than symbolic) attacks. This paper meets this challenge for public-key encryption schemes built from trapdoor permutations and hash functions. Using a novel combination of techniques from computational and symbolic cryptography, we present proof systems for analyzing the chosen-plaintext and chosen-ciphertext security of such schemes in the random oracle model. Building on these proof systems, we develop a toolset that bundles together fully automated proof and attack finding algorithms. We use this toolset to build a comprehensive database of encryption schemes that records attacks against insecure schemes, and proofs with concrete bounds for secure ones.

EasyCrypt is a tool-assisted framework for reasoning about probabilistic computations in the presence of adversarial code, whose main application has been the verification of security properties of cryptographic constructions in the computational model. We report on a significantly enhanced version of EasyCrypt that accommodates a richer, user-extensible language of probabilistic expressions and, more fundamentally, supports reasoning about approximate forms of program equivalence. This enhanced framework allows us to express a broader range of security properties, that notably include approximate and computational differential privacy. We illustrate the use of the framework by verifying two protocols: a two-party protocol for computing the Hamming distance between bit-vectors, yielding two-sided privacy guarantees; and a novel, efficient, and privacy-friendly distributed protocol to aggregate smart meter readings into statistics and bills.

Existing pattern-based compiler technology is unable to effectively exploit the full potential of SIMD architectures. We present a new program synthesis based technique for auto-vectorizing performance critical innermost loops. Our synthesis technique is applicable to a wide range of loops, consistently produces performant SIMD code, and generates correctness proofs for the output code. The synthesis technique, which leverages existing work on relational verification methods, is a novel combination of deductive loop restructuring, synthesis condition generation and a new inductive synthesis algorithm for producing loop-free code fragments. The inductive synthesis algorithm wraps an optimized depth-first exploration of code sequences inside a CEGIS loop. Our technique is able to quickly produce SIMD implementations (up to 9 instructions in 0.12 seconds) for a wide range of fundamental looping structures. The resulting SIMD implementations outperform the original loops by 2.0x-3.7x.

Many cryptographic systems based on elliptic curves are proven secure in the Random Oracle Model, assuming there exist probabilistic functions that map elements in some domain (e.g. bitstrings) onto uniformly and independently distributed points in a curve. When implementing such systems, and in or der for the proof to carry over to the implementation, those mappings must be instantiated with concrete constructions whose behavior does not deviate significantly from random oracles. In contrast to other approaches to public-key cryptography, where candidates to instantiate random oracles have been known for some time, the first generic construction for hashing into ordinary elliptic curves indifferentiable from a random oracle was put forward only recently by Brier et al. We present a machine-checked proof of this construction. The proof is based on an extension of the CertiCrypt framework with logics and mechanized tools for reasoning about approximate forms of observational equivalence, and integrates mathematical libraries of group theory and elliptic curves.

Non-interference guarantees the absence of illicit information flow throughout program execution. It can be enforced by appropriate information flow type systems. Much of the previous work on type systems for non-interference has focused on calculi or high-level programming languages, and existing type systems for low-level languages typically omit objects, exceptions and method calls. We define an information flow type system for a sequential JVM-like language that includes all these programming features, and we prove, in the Coq proof assistant, that it guarantees non-interference. An additional benefit of the formalisation is that we have extracted from our proof a certified lightweight bytecode verifier for information flow. Our work provides, to the best of our knowledge, the first sound and certified information flow type system for such an expressive fragment of the JVM.

Differential privacy is a notion of confidentiality that allows useful computations on sensible data while protecting the privacy of individuals. Proving differential privacy 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 fall out of their scope. Examples include programs that aim for weaker, approximate differential privacy guarantees and programs that achieve differential privacy without using any standard mechanisms. 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 probabilistic relational Hoare logic that enables one to derive differential privacy guarantees for programs from first principles. We demonstrate the applicability of CertiPriv on 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, Gaussian, and exponential mechanisms and of the privacy of randomized and streaming algorithms from the literature.

Developers building cryptography into security-sensitive applications face a daunting task. Not only must they understand the security guarantees delivered by the constructions they choose, they must also implement and combine them correctly and efficiently. Cryptographic compilers free developers from this task by turning high-level specifications of security goals into efficient implementations. Yet, trusting such tools is hard as they rely on complex mathematical machinery and claim security properties that are subtle and difficult to verify. In this paper we present ZKCrypt, an optimizing cryptographic compiler achieving an unprecedented level of assurance without sacrificing practicality for a comprehensive class of cryptographic protocols, known as Zero-Knowledge Proofs of Knowledge. The pipeline of ZKCrypt integrates purpose-built verified compilers and verifying compilers producing formal proofs in the CertiCrypt framework. By combining the guarantees delivered by each stage, ZKCrypt provides assurance that the output implementation securely realizes the abstract proof goal given as input. We report on the main characteristics of ZKCrypt, highlight new definitions and concepts at its foundations, and illustrate its applicability through a representative example of an anonymous credential system

Verified security provides a firm foundation for cryptographic proofs by means of rigorous programming language techniques and verification methods. EasyCrypt is a framework that realizes the verified security paradigm and supports the machine-checked construction and verification of cryptographic proofs using state-of-the-art SMT solvers, automated theorem provers and interactive proof assistants. Previous experiments have shown that EasyCrypt is effective for a posteriori validation of cryptographic systems. In this paper, we report on the first application of verified security to a novel cryptographic construction, with strong security properties and interesting practical features. Specifically, we use EasyCrypt to prove in the Random Oracle Model the IND-CCA security of a redundancy-free public-key encryption scheme based on trapdoor one-way permutations. Somewhat surprisingly, we show that even with a zero-length redundancy, Boneh's SAEP scheme (an OAEP-like construction with a single-round Feistel network rather than two) converts a trapdoor one-way permutation into an IND-CCA-secure scheme, provided the permutation satisfies two additional properties. We then prove that the Rabin function and RSA with short exponent enjoy these properties, and thus can be used to instantiate the construction we propose to obtain efficient encryption schemes. The reduction that justifies the security of our construction is tight enough to achieve practical security with reasonable key sizes.

Cryptographic hash functions provide a basic data authentication mechanism and are used pervasively as building blocks to realize many cryptographic functionalities, including block ciphers, message authentication codes, key exchange protocols, and encryption and digital signature schemes. Since weaknesses in hash functions may imply vulnerabilities in the constructions that build upon them, ensuring their security is essential. Unfortunately, many widely used hash functions, including SHA-1 and MD5, are subject to practical attacks. The search for a secure replacement is one of the most active topics in the field of cryptography. In this paper we report on the first machine-checked and independently-verifiable proofs of collision-resistance and in differentiability of Merkle-Damgaard, a construction that underlies many existing hash functions. Our proofs are built and verified using an extension of the Easy Crypt framework, which relies on state-of-the-art verification tools such as automated theorem provers, SMT solvers, and interactive proof assistants.

Virtualization platforms allow multiple operating systems to run on the same hardware. One of their central goal is to provide strong isolation between guest operating systems, unfortunately, they are often vulnerable to practical side-channel attacks. Cache attacks are a common class of side-channel attacks that use the cache as a side channel. We formalize an idealized model of virtualization that features the cache and the Translation Look aside Buffer (TLB), and that provides an abstract treatment of cache-based side-channels. We then use the model for reasoning about cache-based attacks and countermeasures, and for proving that isolation between guest operating systems can be enforced by flushing the cache upon context switch. In addition, we show that virtualized platforms are transparent, i.e. a guest operating system cannot distinguish whether it executes alone or together with other guest operating systems on the platform. The models and proofs have been machine-checked in the Coqproof assistant.

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.

This series of Workshops is organized by ERCIM (European Research Consortium in Informatics and Mathematics) Working Group on Security and Trust Management, that was established in 2005 to foster collaborative work on all theoretical and practical aspects of security, trust and privacy in ICT within the European research community and to increase co-operation of the research community with the European industry.The four articles in this Special Issue were selected from the 17 papers that were accepted for presentation at the Workshop, out of 40 submissions.The articles cover representative topics of STM, such as access control, privacy, security and trust policies, or formal methods.The paper "Scalable automated symbolic analysis of administrative role-based access control policies by SMT solving" by Armando and Ranise considers the problem of automatically analysing administrative role based access control (ARBAC) policies.Such analyses are extremely useful to understand the subtle implications of complex combinations of authorizations, and more generally to ensure that ARBAC systems are scalable and can evolve gracefully.Authors isolate a new class of properties (user-role reachability) for ARBAC policies and leverage state-of-the-art SMT solvers to check automatically their validity.Finally, the strength of their approach is demonstrated through extensive experimental evaluation.In their paper "Stateful authorization logic -proof theory and a case study", Garg and Pfenning embark on the proof-theoretical study of logics for stateful authorization policies; that is, policies that depend on externally controlled conditions.They introduce an expressive logic, called BL, for reasoning about stateful authorization policies.The expressiveness of the logic is achieved through the adoption of a modal connective that allows delegation, and through an explicit modeling of time.Thanks to a careful design that crisply separates between the validity of state predicates and BL judgments, the authors provide an elegant treatment of the meta-theory of BL, and prove that it enjoys admissibility of cut, an essential property that entails consistency.The BL logic is empirically validated through a detailed case study of the US authorization policies for accessing to sensitive intelligence information.The paper "Modeling and preventing inferences from sensitive value distributions in data release" by Bezzi, De Capitani di Vimercati, Foresti, Livraga, Samarati and Sassi considers the following setting: assume a database contains many entries, each of which in its own is considered non-sensitive and can be declassified, but some

There are two active and independent lines of research that aim at quantifying the amount of information that is disclosed by computing on confidential data. Each line of research has developed its own notion of confidentiality: on the one hand, differential privacy is the emerging consensus guarantee used for privacy-preserving data analysis. On the other hand, information-theoretic notions of leakage are used for characterizing the confidentiality properties of programs in language-based settings. The purpose of this article is to establish formal connections between both notions of confidentiality, and to compare them in terms of the security guarantees they deliver. We obtain the following results. First, we establish upper bounds for the leakage of every ϵ-differentially private mechanism in terms of eps and the size of the mechanism's input domain. We achieve this by identifying and leveraging a connection to coding theory. Second, we construct a class of ϵ-differentially private channels whose leakage grows with the size of their input domains. Using these channels, we show that there cannot be domain-size-independent bounds for the leakage of all ϵ-differentially private mechanisms. Moreover, we perform an empirical evaluation that shows that the leakage of these channels almost matches our theoretical upper bounds, demonstrating the accuracy of these bounds. Finally, we show that the question of providing optimal upper bounds for the leakage of ϵ-differentially private mechanisms in terms of rational functions of ϵ is in fact decidable.

Information flow policies are confidentiality policies that control information leakage through program execution. A common way to enforce secure information flow is through information flow type systems. Although type systems are compositional and usually enjoy decidable type checking or inference, their extensibility is very poor: type systems need to be redefined and proved sound for each new variation of security policy and programming language for which secure information flow verification is desired. In contrast, program logics offer a general mechanism for enforcing a variety of safety policies, and for this reason are favoured in Proof Carrying Code, which is a promising security architecture for mobile code. However, the encoding of information flow policies in program logics is not straightforward because they refer to a relation between two program executions. The purpose of this paper is to investigate logical formulations of secure information flow based on the idea of self-composition, which reduces the problem of secure information flow of a program P to a safety property for a program derived from P by composing P with a renaming of itself. Self-composition enables the use of standard techniques for information flow policy verification, such as program logics and model checking, that are suitable in Proof Carrying Code infrastructures. We illustrate the applicability of self-composition in several settings, including different security policies such as non-interference and controlled forms of declassification, and programming languages including an imperative language with parallel composition, a non-deterministic language and, finally, a language with shared mutable data structures.

A certificate is a mathematical object that can be used to establish that a piece of mobile code satisfies some security policy. In general, certificates cannot be generated automatically. There is thus an interest in developing methods to reuse certificates generated for source code to provide strong guarantees of the compiled code correctness. Certificate translation is a method to transform certificates of program correctness along semantically justified program transformations. These methods have been developed in previous work, but they were strongly dependent on particular programming and verification settings. This article provides a more general development in the setting of abstract interpretation, showing the scalability of certificate translation.

Computational Indistinguishability Logic (CIL) is a logic for reasoning about cryptographic primitives in computational models. It captures reasoning patterns that are common in provable security, such as simulations and reductions. CIL is sound for the standard model, but also supports reasoning in the random oracle and other idealized models.

Zero-knowledge proofs have a vast applicability in the domain of cryptography, stemming from the fact that they can be used to force potentially malicious parties to abide by the rules of a protocol, without forcing them to reveal their secrets. Σ-protocols are a class of zero-knowledge proofs that can be implemented efficiently and that suffice for a great variety of practical applications. This paper presents a first machine-checked formalization of a comprehensive theory of Σ-protocols. The development includes basic definitions, relations between different security properties that appear in the literature, and general composability theorems. We show its usefulness by formalizing—and proving the security—of concrete instances of several well-known protocols. The formalization builds on CertiCrypt, a framework that provides support to reason about cryptographic systems in the Coq proof assistant, and that has been previously used to formalize security proofs of encryption and signature schemes.

Anonymous communication protocols must achieve two seemingly contradictory goals: privacy (informally, they must guarantee the anonymity of the parties that send/receive information), and robustness (informally, they must ensure that the messages are not tampered). However, the long line of research that defines and analyzes the security of such mechanisms focuses almost exclusively on the former property and ignores the latter. In this paper, we initiate a rigorous study of robustness properties for anonymity protocols. We identify and formally define, using the style of modern cryptography, two related but distinct flavors of robustness. Our definitions are general (e.g. they strictly generalize the few existent notions for particular protocols) and flexible (e.g. they can be easily adapted to purely combinatorial/probabilistic mechanisms). We demonstrate the use of our definitions through the analysis of several anonymity mechanisms (Crowds, broadcast-based mix-nets, DC-nets, Tor). Notably, we analyze the robustness of a protocol by Golle and Juels for the dining cryptographers problem, identify a robustness-related weakness of the protocol, and propose and analyze a stronger version.

End-to-End security of mobile code requires that the code neither intentionally nor accidentally propagates sensitive information to an adversary. Although mobile code is commonly multithreaded low-level code, there lack enforcement mechanisms that ensure information security for such programs. The modularity is three-fold: we give modular extensions of sequential semantics, sequential security typing, and sequential security-type preserving compilation that allow us enforcing security for multithreaded programs. Thanks to the modularity, there are no more restrictions on multithreaded source programs than on sequential ones, and yet we guarantee that their compilations are provably secure for a wide class of schedulers.

As cryptographic proofs have become essentially unverifiable, cryptographers have argued in favor of developing techniques that help tame the complexity of their proofs. Game-based techniques provide a popular approach in which proofs are structured as sequences of games and in which proof steps establish the validity of transitions between successive games. Code-based techniques form an instance of this approach that takes a code-centric view of games, and that relies on programming language theory to justify proof steps. While code-based techniques contribute to formalize the security statements precisely and to carry out proofs systematically, typical proofs are so long and involved that formal verification is necessary to achieve a high degree of confidence. We present Certicrypt, a framework that enables the machine-checked construction and verification of code-based proofs. Certicrypt is built upon the general-purpose proof assistant Coq, and draws on many areas, including probability, complexity, algebra, and semantics of programming languages. Certicrypt provides certified tools to reason about the equivalence of probabilistic programs, including a relational Hoare logic, a theory of observational equivalence, verified program transformations, and game-based techniques such as reasoning about failure events. The usefulness of Certicrypt is demonstrated through various examples, including a proof of semantic security of OAEP (with a bound that improves upon existing published results), and a proof of existential unforgeability of FDH signatures. Our work provides a first yet significant step towards Halevi's ambitious programme of providing tool support for cryptographic proofs.

We present two machine-checked proofs of the existentialunforgeability under adaptive chosen-message attacks of the FullDomain Hash signature scheme. These proofs formalize the originalargument of Bellare and Rogaway, and an optimal reduction by Coronthat provides a tighter bound on the probability of a forgery. Bothproofs are developed using CertiCrypt, a general framework toformalize exact security proofs of cryptographic systems in thecomputational model. Since CertiCrypt is implemented on top of theCoq proof assistant, the proofs are highly trustworthy and can beverified independently and fully automatically.

Proof Carrying Code provides trust in mobile code by requiring certificates that ensure the code adherence to specific conditions. The prominent approach to generate certificates for compiled code is Certifying Compilation, that automatically generates certificates for simple safety properties. In this work, we present Certificate Translation, a novel extension for standard compilers that automatically transforms formal proofs for more expressive and complex properties of the source program to certificates for the compiled code. The article outlines the principles of certificate translation, instantiated for a nonoptimizing compiler and for standard compiler optimizations in the context of an intermediate RTL Language.

Formalizing appropriate information policies that authorize some controlled form of information release, and providing sound analyses for these policies is a necessary step towards practical applications of language-based security. We propose a modular method to enhance non-interference type systems to support controlled forms of information release that combine the what and where dimensions of declassification. As a case study, we derive from earlier work on non-interference type systems new type systems that soundly enforce declassification policies for sequential fragments of the Java Virtual Machine. Our work provides the first modular method to define sound type systems for declassification policies, and the first instance of a sound type system that supports declassification policies for unstructured languages.

Program verification environments increasingly rely on hybrid methods that combine static analyses and verification condition generation. While such verification environments operate on source programs, it is often preferable to achieve guarantees about executable code. We show that, for a hybrid verification method based on numerical static analysis and verification condition generation, compilation preserves proof obligations and therefore it is possible to transfer evidence from source to compiled programs. Our result relies on the preservation of the solutions of analysis by compilation; this is achieved by relying on a byte code analysis that performs symbolic execution of stack expressions in order to overcome the loss of precision incurred by performing static analyses on compiled (rather than source) code. Finally, we show that hybrid verification methods are sound by proving that every program provable by hybrid methods is also provable (at a higher cost) by standard methods.

Information flow type systems provide an elegant means to enforce confidentiality of programs. Using the proof assistant Isabelle/HOL, we have specified an information flow type system for a concurrent language featuring primitives for scheduling, and shown that typable programs are non-interfering for a possibilistic notion of non-interference. The development, which constitutes to our best knowledge the first machine-checked account of non-interference for a concurrent language, takes advantage of the proof assistant facilities to structure the proofs about different views of the programming language and to identify the relationships among them and the type system. Our language and type system generalize previous work of Boudol and Castellani, Theoretical Computer Science 281 (2002), 109–130, in particular by including arrays and lifting several convenient but unnecessary conditions in the syntax and type system of the work of Boudol and Castellani. We illustrate the generality of our language and the usefulness of our type system with a medium size example.

Language-based security provides a means to enforce end-to-end confidentiality and integrity policies in mobile code scenarios, and is increasingly being contemplated by the smart-card and mobile phone industry as a solution to enforce information flow and resource control policies. Two threads of work have emerged in research on language-based security: work that focuses on enforcing security policies for source code, which is tailored towards developers that want to increase confidence in their applications, and work that focuses on efficiently verifying similar policies for byte-code, which is tailored to code consumers that want to protect themselves against hostile applications. These lines of work serve different purposes - and thus have been developed independently - but connecting them is a key step towards the deployment of language-based security in practical applications. This paper introduces a systematic technique to connect source code and bytecode security type systems. The technique is applied to an information flow type system for a fragment of Java with exceptions, thus confronting challenges in both control and data flow tracking

Pure Type Systems (PTS) come in two flavours: domain-free systems with untyped $\lambda$ -abstractions (i.e. of the form $\lambda{x}.{M}$ ); and domain-free systems with typed $\lambda$ -abstractions (i.e. of the form $\lambda{x}{A}{M}$ ). Both flavours of systems are related by an erasure function $\er{.}$ that removes types from $\lambda$ -abstractions. Preservation of Equational Theory , which states the equational theories of both systems coincide through the erasure function, is a property of functional and normalizing PTSs. In this paper we establish that Preservation of Equational Theory fails for some non-normalizing PTSs, including the PTS with $\ast:\ast$. The gist of our argument is to exhibit a typable expression $Y_H$ whose erasure $\er{Y}$ is a fixpoint combinator, but which is not a fixpoint combinator itself.

Memory consumption policies provide a means to control resource usage on constrained devices, and play an important role in ensuring the overall quality of software systems, and in particular resistance against resource exhaustion attacks. Such memory consumption policies have been previously enforced through static analysis, which yield automatic bounds at the cost of precision, or run-time analysis, which incur an overhead that is not acceptable for constrained devices. In this paper, we study the use of logical methods to specify and statically verify precise memory consumption policies for Java bytecode programs. First, we demonstrate how the bytecode specification language (a variant of the Java modelling language tailored to bytecode) can be used to specify precise memory consumption policies for (sequential) Java applets, and how verification tools can be used to enforce such memory consumption policies. Second, we consider the issue of inferring some of the annotations required to express the memory consumption policy, and report on an inference algorithm. Our broad conclusion is that logical methods provide a suitable means to specify and verify expressive memory consumption policies, with an acceptable overhead.

Type isomorphisms are pairs of functions are mutually inverse. In principle, type isomorphisms, viewed as back-and-forth coercions, could be used in the context of proof assistants based on dependent type theory in order to relate equivalent representations of mathematical notions, for example, the polar and cartesian representations of complex numbers. However, the coercions that map one representation into another are not mutually inverse because of the intensional nature of dependent type theories. Consequently, the standard concept of type isomorphisms has limited applicability in the context of proof assistants.In order to circumvent this problem, we develop a computational interpretation of implicit coercions that allows for the definition of back-and-forth coercions without requiring them to be mutually inverse. We illustrate the usefulness of our approach in a number of formal developments that require us to navigate between different representations of mathematical objects or structures. We also discuss important meta-theoretical properties of our interpretation.

Modern programming languages rely on advanced type systems that detect errors at compile-time. While the benefits of type systems have long been recognized, there are some areas where the standard systems in programming languages are not expressive enough. Language designers usually trade expressiveness for decidability of the type system. Some interesting programs will always be rejected (despite their semantical soundness) or be assigned uninformative types.

This paper introduces $\lambda^\widehat$ , a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalisation of typable terms and to be stronger than a related system $\lambda_{\mathcal{G}}$ in which termination is ensured by a syntactic guard condition. The system can, at will, be extended to support coinductive types and corecursive function definitions also.

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Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.

We investigate CPS translatability of typed λ-calculi with inductive and coinductive types. We show that tenable Plotkin-style call-by-name CPS translations exist for simply typed λ-calculi with a natural number type and stream types and, more generally, with arbitrary positive inductive and coinductive types. These translations also work in the presence of control operators and generalize for dependently typed calculi where case-like eliminations are only allowed in non-dependent forms. No translation is possible along the same lines for small Σ-types and sum types with dependent case.

Pure type systems make use of domain-full λ-abstractions λx:D.M. We present a variant of pure type systems, which we call domain-free pure type systems, with domain-free λ-abstractions λx.M. Domain-free pure type systems have a number of advantages over both pure type systems and so-called type assignment systems (they also have some disadvantages), and have been used in theoretical developments as well as in implementations of proof-assistants. We study the basic properties of domain-free pure type systems, establish their formal relationship with pure type systems and type assignment systems, and give a number of applications of these correspondences.

Injective pure type systems form a large class of pure type systems for which one can compute by purely syntactic means two sorts elmt(Γ[mid ] M ) and sort(Γ[mid ] M ), where Γ is a pseudo-context and M is a pseudo-term, and such that for every sort s , formula here By eliminating the problematic clause in the (abstraction) rule in favor of constraints over elmt(.[mid ].) and sort(.[mid ].), we provide a sound and complete type-checking algorithm for injective pure type systems. In addition, we prove expansion postponement for a variant of injective pure type systems where the problematic clause in the (abstraction) rule is replaced in favor of constraints over elmt(.[mid ].) and sort(.[mid ].).