Quantum protocols such as the BB84 Quantum Key Distribution protocol exchange qubits to achieve information-theoretic security guarantees. Many variants thereof were proposed, some of them being already deployed. Existing security proofs in that field are mostly tedious, error-prone pen-and-paper proofs of the core protocol only that rarely account for other crucial components such as authentication. This calls for formal and automated verification techniques that exhaustively explore all possible intruder behaviors and that scale well. The symbolic approach offers rigorous, mathematical frameworks and automated tools to analyze security protocols. Based on well-designed abstractions, it has allowed for large-scale formal analyses of real-life protocols such as TLS 1.3 and mobile telephony protocols. Hence a natural question is: Can we use this successful line of work to analyze quantum protocols? This paper proposes a first positive answer and motivates further research on this unexplored path.
Quantum protocols such as the BB84 Quantum Key Distribution protocol exchange qubits to achieve information-theoretic security guarantees. Many variants thereof were proposed, some of them being already deployed. Existing security proofs in that field are mostly tedious, error-prone pen-and-paper proofs of the core protocol only that rarely account for other crucial components such as authentication. This calls for formal and automated verification techniques that exhaustively explore all possible intruder behaviors and that scale well. The symbolic approach offers rigorous, mathematical frameworks and automated tools to analyze security protocols. Based on well-designed abstractions, it has allowed for large-scale formal analyses of real-life protocols such as TLS 1.3 and mobile telephony protocols. Hence a natural question is: Can we use this successful line of work to analyze quantum protocols? This paper proposes a first positive answer and motivates further research on this unexplored path.
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.
Our objective in this thesis is to perform formal specification and automated validation of the class of protocols associated with quantum cryptography. Current analyses and proofs regarding quantum cryptographic protocols and their security are information-theoretic and also require in--depth understanding of the underlying physics. The alternative approach presented here involves the use of computer modelling languages and verification software. In particular, the logical model checker SPIN and the probabilistic model checker PRISM are used to analyse quantum key distribution, entanglement and quantum teleportation. The value of our approach lies not only in the fact that it is conceptually simpler than existing proofs, but in that it allows us to disambiguate protocol definitions and to assess their properties in various circumstances. By varying the values of certain parameters, different attacks on the protocols can be attempted and implementation-specific attributes can be analysed. The quantum key distribution protocol BB84 has been proved to be unconditionally secure against all types of eavesdropping. Although a general proof of this result is extremely hard to generate automatically, it is possible to develop specific protocol attacks and validate the protocol by computer. Assuming two types of eavesdropping attack (intercept-resend and random-substitute), we have used the PRISM tool to compute exactly the probability of successful eavesdropping, and found it to diminish as the number of transmitted qubits is increased. This result is linked to Mayers' security criteria for BB84. Also, we briefly describe a new quantum protocol definition language, named qSpec. The recent CQP process algebra, due to Nagarajan and Gay, is reviewed and examples of its application are provided. Alternative formalisms and validation tools are discussed, including QPAlg, PROBMELA and probUSM, the logic of knowledge and time for quantum systems due to Van der Meyden, and the QCSim simulator.
We propose to analyse quantum protocols by applying formal verification techniques developed in classical computing for the analysis of communicating concurrent systems. One area of successful application of these techniques is that of classical security protocols, exemplified by Lowe's discovery and fix of a flaw in the well-known Needham-Schroeder authentication protocol. Secure quantum cryptographic protocols are also notoriously difficult to design. Quantum cryptography is therefore an interesting target for formal verification, and provides our first example; we expect the approach to be transferable to more general quantum information processing scenarios. The example we use is the quantum key distribution protocol proposed by Bennett and Brassard, commonly referred to as BB84. We present a model of the protocol in the process calculus CCS and the results of some initial analyses using the Concurrency Workbench of the New Century (CWB-NC).
This paper discusses the use of computer-aided verification as a practical means for analysing quantum information systems; specifically, the BB84 protocol for quantum key distribution is examined using this method. This protocol has been shown to be unconditionally secure against all attacks in an information-theoretic setting, but the relevant security proof requires a thorough understanding of the formalism of quantum mechanics and is not easily adaptable to practical scenarios. Our approach is based on probabilistic model-checking; we have used the PRISM model-checker to show that, as the number of qubits transmitted in BB84 is increased, the equivocation of the eavesdropper with respect to the channel decreases exponentially. We have also shown that the probability of detecting the presence of an eavesdropper increases exponentially with the number of qubits. The results presented here are a testament to the effectiveness of the model-checking approach for systems where analytical solutions may not be possible or plausible.
We present a rigorous and complete security proof of the decoy-state BB84 quantum key distribution (QKD) protocol. Our analysis aims to achieve a high standard of mathematical rigour and completeness, thereby providing the necessary foundation for certification and standardization efforts. Beyond establishing the security of a specific protocol, this work develops a general and modular framework that can be readily adapted to a broad class of QKD protocols, including both prepare-and-measure and entanglement-based variants. Our framework unifies all major ingredients required for the analysis of realistic QKD protocols, including the analysis of classical authentication and classical processing, source-replacement schemes, finite-size analysis, source maps, squashing maps, and decoy-state techniques. In doing so, this work consolidates a diverse range of techniques scattered across the QKD literature into a unified formalism, representing a general and rigorous treatment of QKD security. Finally, it outlines a clear path towards incorporating practical imperfections within the same framework, thereby laying the groundwork for addressing implementation security in future analysis.
Quantum Key Distribution (QKD) promises information-theoretic security, yet practical implementations ranging from point-to-point links to large-scale networks can introduce protocol and architectural vulnerabilities that undermine this guarantee. We present a formal verification framework based on QTPi, a process calculus for distributed quantum systems, to model and verify QKD protocols at the Quantum Link Layer (QLL). Using BB84 as a case study, we specify concurrent processes for Alice, Bob, and supporting components, and formally check functional requirements defined within the Innovate UK project QAssure, such as key agreement (QLL-EM-1). Our approach integrates symbolic simulation and formal assertions to automatically verify that critical QLL properties hold across executions. This establishes a systematic methodology for providing assurance of QKD protocols in realistic settings and lays the foundation for future quantitative verification of probabilistic security guarantees, including resilience to eavesdropping and bounded key failure rates.