Reducing Equational Theories for the Decision of Static Equivalence

International audienceStatic equivalence is a well established notion of indistinguishability of sequences of terms which is useful in the symbolic analysis of cryptographic protocols. Static equivalence modulo equational theories allows for a more accurate representation of cryptographic primitives by modelling properties of operators by equational axioms. We develop a method that allows us in some cases to simplify the task of deciding static equivalence in a multi-sorted setting, by removing a symbol from the term signature and reducing the problem to several simpler equational theories. We illustrate our technique at hand of bilinear pairings

YAPA: A generic tool for computing intruder knowledge

Reasoning about the knowledge of an attacker is a necessary step in many formal analyses of security protocols. In the framework of the applied pi calculus, as in similar languages based on equational logics, knowledge is typically expressed by two relations: deducibility and static equivalence. Several decision procedures have been proposed for these relations under a variety of equational theories. However, each theory has its particular algorithm, and none has been implemented so far. We provide a generic procedure for deducibility and static equivalence that takes as input any convergent rewrite system. We show that our algorithm covers most of the existing decision procedures for convergent theories. We also provide an efficient implementation, and compare it briefly with the tools ProVerif and KiSs

Deciding equivalence-based properties using constraint solving

Formal methods have proved their usefulness for analyzing the security of protocols. Most existing results focus on trace properties like secrecy or authentication. There are however several security properties, which cannot be defined (or cannot be naturally defined) as trace properties and require a notion of behavioural equivalence. Typical examples are anonymity, privacy related properties or statements closer to security properties used in cryptography. In this paper, we consider three notions of equivalence defined in the applied pi calculus: observational equivalence, may-testing equivalence, and trace equivalence. First, we study the relationship between these three notions. We show that for determinate processes, observational equivalence actually coincides with trace equivalence, a notion simpler to reason with. We exhibit a large class of determinate processes, called simple processes, that capture most existing protocols and cryptographic primitives. While trace equivalence and may-testing equivalence seem very similar, we show that may-testing equivalence is actually strictly stronger than trace equivalence. We prove that the two notions coincide for image-finite processes, such as processes without replication. Second, we reduce the decidability of trace equivalence (for finite processes) to deciding symbolic equivalence between sets of constraint systems. For simple processes without replication and with trivial else branches, it turns out that it is actually sufficient to decide symbolic equivalence between pairs of positive constraint systems. Thanks to this reduction and relying on a result first proved by M. Baudet, this yields the first decidability result of observational equivalence for a general class of equational theories (for processes without else branch nor replication). Moreover, based on another decidability result for deciding equivalence between sets of constraint systems, we get decidability of trace equivalence for processes with else branch for standard primitives

Formal Models and Techniques for Analyzing Security Protocols: A Tutorial

International audienceSecurity protocols are distributed programs that aim at securing communications by the means of cryptography. They are for instance used to secure electronic payments, home banking and more recently electronic elections. Given The financial and societal impact in case of failure, and the long history of design flaws in such protocol, formal verification is a necessity. A major difference from other safety critical systems is that the properties of security protocols must hold in the presence of an arbitrary adversary. The aim of this paper is to provide a tutorial to some modern approaches for formally modeling protocols, their goals and automatically verifying them

New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized

The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation fails to identify two `obviously' equal terms. If only the machine could decide a richer theory! We propose a way to decide theories which supplement evaluation with `$\nu$-rules', rearranging the neutral parts of normal forms, and report a successful initial experiment. We study a simple -calculus with primitive fold, map and append operations on lists and develop in Agda a sound and complete decision procedure for an equational theory enriched with monoid, functor and fusion laws

A system for computational analysis and reconstruction of 3D comminuted bone fractures

High energy impacts at joint locations often generate highly fragmented, or comminuted bone fractures. A leading current approach for treatment requires physicians qualitatively to classify the fracture to one of four possible fracture severity cases. Each case then has a sequence of best-practices for obtaining the best possible prognosis for the patient. It has been observed that qualitative evaluation of fracture severity by physicians can vary significantly which can lead to potential mis-classification and mis-treatment of these fracture cases. Major indicators of fracture severity are (i) fracture surface area, i.e., how much surface area was generated when the bone broke apart and (ii) dispersion, i.e., how far the fragments have rotated and translated from their original anatomic positions. Work in this dissertation develops computational tools that solve the bone puzzle-solving problem automatically or semi-automatically and extract previously unavailable quantitative information for these indicators from each bone fragment that are intended to assist physicians in making a more accurate and reliable fracture severity classification. The system applies novel three-dimensional (3D) puzzle-solving algorithms to identify the fracture fragments in the CT image data and piece them back together in a virtual environment. Doing so provides quantitative values for both fracture surface area and dispersion that reduce variability in fracture severity classifications and prevent mis-diagnosis for fracture cases that may be difficult to qualitatively classify using traditional approaches. This dissertation describes the system, the underlying algorithms and demonstrates the virtual reconstruction results and quantitative analysis of comminuted bone fractures from six clinical cases

Reasoning about recognizability in security protocols

Although verifying a message has long been recognized as an important concept, which has been used explicitly or implicitly in security protocol analysis, there is no consensus on its exact meaning. Such a lack of formal treatment of the concept makes it extremely difficult to evaluate the vulnerability of security protocols. This dissertation offers a precise answer to the question: What is meant by saying that a message can be "verified''? The core technical innovation is a third notion of knowledge in security protocols -- recognizability. It can be considered as intermediate between deduction and static equivalence, two classical knowledge notions in security protocols. We believe that the notion of recognizability sheds important lights on the study of security protocols. More specifically, this thesis makes four contributions. First, we develop a knowledge model to capture an agent's cognitive ability to understand messages. Thanks to a clear distinction between de re/dicto interpretations of a message, the knowledge model unifies both computational and symbolic views of cryptography gracefully. Second, we propose a new notion of knowledge in security protocols -- recognizability -- to fully capture one's ability or inability to cope with potentially ambiguous messages. A terminating procedure is given to decide recognizability under the standard Dolev-Yao model. Third, we establish a faithful view of the attacker based on recognizability. This yields new insights into protocol compilations and protocol implementations. Specifically, we identify two types of attacks that can be thawed through adjusting the protocol implementation; and show that an ideal implementation that corresponds to the intended protocol semantics does not always exist. Overall, the obtained attacker's view provides a path to more secure protocol designs and implementations. Fourth, we use recognizability to provide a new perspective on type-flaw attacks. Unlike most previous approaches that have focused on heuristic schemes to detect or prevent type-flaw attacks, our approach exposes the enabling factors of such attacks. Similarly, we apply the notion of recognizability to analyze off-line guessing attacks. Without enumerating rules to determine whether a guess can be "verified'', we derive a new definition based on recognizability to fully capture the attacker's guessing capabilities. This definition offers a general framework to reason about guessing attacks in a symbolic setting, independent of specific intruder models. We show how the framework can be used to analyze both passive and active guessing attacks

Knowledge Problems in Security Protocols: Going Beyond Subterm Convergent Theories

We introduce a new form of restricted term rewrite system, the graph-embedded term rewrite system. These systems, and thus the name, are inspired by the graph minor relation and are more flexible extensions of the well-known homeomorphic-embedded property of term rewrite systems. As a motivating application area, we consider the symbolic analysis of security protocols, and more precisely the two knowledge problems defined by the deduction problem and the static equivalence problem. In this field restricted term rewrite systems, such as subterm convergent ones, have proven useful since the knowledge problems are decidable for such systems. However, many of the same decision procedures still work for examples of systems which are "beyond subterm convergent". However, the applicability of the corresponding decision procedures to these examples must often be proven on an individual basis. This is due to the problem that they don\u27t fit into an existing syntactic definition for which the procedures are known to work. Here we show that many of these systems belong to a particular subclass of graph-embedded convergent systems, called contracting convergent systems. On the one hand, we show that the knowledge problems are decidable for the subclass of contracting convergent systems. On the other hand, we show that the knowledge problems are undecidable for the class of graph-embedded systems

Automated Verification of Equivalence Properties of Cryptographic Protocols

Indistinguishability properties are essential in formal verification of cryptographic protocols. They are needed to model anonymity properties, strong versions of confidentiality and resistance against offline guessing attacks, which can be conveniently modeled using process equivalences. We present a novel procedure to verify equivalence properties for a bounded number of sessions of cryptographic protocols. As in the applied pi-calculus, our protocol specification language is parametrized by a first-order sorted term signature and an equational theory which allows formalization of algebraic properties of cryptographic primitives. Our procedure is able to verify trace equivalence for determi-nate cryptographic protocols. On determinate protocols, trace equivalence coincides with observational equivalence which can therefore be automatically verified for such processes. When protocols are not determinate our procedure can be used for both under-and over-approximations of trace equivalence, which proved successful on examples. The procedure can handle a large set of cryptographic primitives, namely those that can be modeled by an optimally reducing convergent rewrite system. The procedure is based on a fully abstract modelling of the traces of a bounded number of sessions of the protocols into first-order Horn clauses on which a dedicated resolution procedure is used to decide equivalence properties. We have shown that our procedure terminates for the class of subterm convergent equational theories. Moreover, the procedure has been implemented in a prototype tool A-KiSs (Active Knowledge in Security Protocols) and has been effectively tested on examples. Some of the examples were outside the scope of existing tools, including checking anonymity of an electronic voting protocol