Abstract One important application of quantum process algebras is to formally verify quantum communication protocols. With a suitable notion of behavioural equivalence and a decision method, one can determine if an implementation of a protocol is consistent with its specification. Ground bisimulation is a convenient behavioural equivalence for quantum processes because of its associated coinduction proof technique. We exploit this technique to design and implement two on-the-fly algorithms for the strong and weak versions of ground bisimulation to check if two given processes in quantum CCS are equivalent. We then develop a tool that can verify interesting quantum protocols such as the BB84 quantum key distribution scheme.
Abstract One important application of quantum process algebras is to formally verify quantum communication protocols. With a suitable notion of behavioural equivalence and a decision method, one can determine if an implementation of a protocol is consistent with its specification. Ground bisimulation is a convenient behavioural equivalence for quantum processes because of its associated coinduction proof technique. We exploit this technique to design and implement two on-the-fly algorithms for the strong and weak versions of ground bisimulation to check if two given processes in quantum CCS are equivalent. We then develop a tool that can verify interesting quantum protocols such as the BB84 quantum key distribution scheme.
Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit more faults than classical protocol designers since human intuition is poorly adapted to the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. An important issue in quantum process algebra is to discover a quantum generalization of bisimulation preserved by various process constructs, in particular, parallel composition, where one of the major differences between classical and quantum systems, namely quantum entanglement, is present. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communication is involved. Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communication) scheme, where classical communication must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communication is crucial for process algebra approach to verification of quantum cryptographic protocols. In this article we introduce novel notions of strong bisimulation and weak bisimulation for quantum processes, and prove that they are congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communication are present. We also establish some basic algebraic laws for these bisimulations. In particular, we show the uniqueness of the solutions to recursive equations of quantum processes, which proves useful in verifying complex quantum protocols. To capture the idea that a quantum process approximately implements its specification, and provide techniques and tools for approximate reasoning, a quantified version of strong bisimulation, which defines for each pair of quantum processes a bisimulation-based distance characterizing the extent to which they are strongly bisimilar, is also introduced.
t is recognized that security verification of cryptographic protocols tends to be difficult and in fact, some flaws on protocol designs or security proofs were found after they had been presented. The use of formal methods is a way to deal with such complexity. Especially, process calculi are suitable to describe parallel systems. Bisimilarity, which denotes that two processes behave indistinguishably from the outside, is a key notion in process calculi. However, by-hand verification of bisimilarity is often tedious when the processes have many long branches in their transitions. We developed a software tool to automatically verify bisimulation relation in a quantum process calculus qCCS and applied it to Shor and Preskill's security proof of BB84. The tool succeeds to verify the equivalence of BB84 and an EDP-based protocol, which are discussed in their proof.
Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit much more faults than classical protocol designers since human intuition is much better adapted to the classical world than the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. One of the most serious issues in quantum process algebra is to discover a quantum generalization of the notion of bisimulation, which lies in a central position in process algebra, preserved by parallel composition in the presence of quantum entanglement, which has no counterpart in classical computation. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communications are involved. Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communications) scheme, where classical communications must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communications is crucial for process algebra approach to verification of quantum cryptographic protocols. In this paper we introduce a novel notion of bisimulation for quantum processes and prove that it is congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communications are present. We also establish some basic algebraic laws for this bisimulation. In particular, we prove uniqueness of the solutions to recursive equations of quantum processes, which provides a powerful proof technique for verifying complex quantum protocols.
Bisimulation is a fundamental concept in the classical concurrency theory for comparing the behaviour of nondeterministic processes. It admits elegant characterisations from various perspectives such as fixed point theory, modal logics, game theory, coalgebras etc. In this paper, we review some key ideas used in the formulations and characterisations of reasonable notions of bisimulations for both probabilistic and quantum processes. To some extent the transition from probabilistic to quantum concurrency theory is smooth and natural. However, new ideas need also to be introduced. We have not yet reached the stage of formally verifying quantum communication protocols and quantum algorithms using bisimulations implemented by automatic tools. We discuss some recent efforts in this direction.
With the previous notions of bisimulation presented in the literature, to check if two quantum processes are bisimilar, we have to instantiate their free quantum variables with arbitrary quantum states, and verify the bisimilarity of the resulting configurations. This makes checking bisimilarity infeasible from an algorithmic point of view, because quantum states constitute a continuum. In this article, we introduce a symbolic operational semantics for quantum processes directly at the quantum operation level, which allows us to describe the bisimulation between quantum processes without resorting to quantum states. We show that the symbolic bisimulation defined here is equivalent to the open bisimulation for quantum processes in previous work, when strong bisimulations are considered. An algorithm for checking symbolic ground bisimilarity is presented. We also give a modal characterisation for quantum bisimilarity based on an extension of Hennessy-Milner logic to quantum processes.
In general, it is difficult to verify security of cryptographic protocols.Indeed, flaws of designs and security proofs of some protocols were found after they had been presented.In deductive verification using formal methods, protocols and security properties are described in formal languages, and correctness of designs and security proofs are deduced by inference rules.While a number of formal frameworks and verification tools have been developed and applied to classical protocols, few formal methods have been applied to security proofs of quantum protocols.The contributions of this thesis consist of the following three results.First, we developed a software tool to verify bisimilarity of configurations (pairs of processes and quantum states) of qCCS, a quantum process calculus presented by Feng et al.Bisimilar configurations behave indistinguishably from the outside.We designed a formal framework for the verifier, which we call nondeterministic qCCS, on the basis of qCCS by Feng et al.While the transition system of qCCS is both nondeterministic and probabilistic, we presented a nondeterministic and non-probabilistic transition system for configurations, extending the definition of them.This allows the verifier to verify bisimilarity efficiently.Next, we designed the verifier to handle security parameters and quantum states symbolically.A purpose is to apply it to quantum cryptographic protocols, where the dimensions of quantum states depend on security parameters.When the verifier checks bisimulation relation of configurations, it uses user-defined equations on the symbolic representations to check the quantum states that the outsider can access are always equal.Besides, the verifier is sound with respect to qCCS, that is, when it runs with two configurations as input and returns true, a symbol representing success of the verification, the configurations are in bisimulation relation in the qCCS's definition.Second, we defined a notion of approximate bisimulation relation on configurations of nondeterministic qCCS, and extended the verifier to check the approximate bisimilarity.Approximately bisimilar configurations reveal quantum states with close trace distance to the outsider after transitions.This property is useful in security proofs of quantum key distribution protocols.We then proved the approximate bisimulation relation is closed under application of evaluation contexts of processes.Third, we formally verified Shor and Preskill's security proof of BB84 quantum key distribution protocol.They first considered another protocol (the EDP-based protocol) and proved the security of BB84 and the EDP-based protocol is equivalent.They next proved the latter is secure.For the first step of their proof, we formalized the two protocols as configurations and formally verified that they are bisimilar.For the second step, we defined a completely secure protocol (EDPideal) and formally verified that the configurations of it and the EDP-based protocol are approximately bisimilar.This is the first work where a security proof of a quantum cryptographic protocol is mechanically verified using a software tool.