[Paper Review] An Automated Analysis of the Security of Quantum Key Distribution

Rajagopal Nagarajan, Nickolas Papanikolaou+2 more|ArXiv.org|Feb 9, 2005

Quantum Computing Algorithms and Architecture20 references19 citations

TL;DR

This paper applies probabilistic model checking using PRISM to formally verify the security of the BB84 quantum key distribution protocol. It demonstrates that the probability of eavesdropping detection and the eavesdropper's information gain decrease exponentially with the number of transmitted qubits, confirming the protocol's unconditional security in a practical, implementable setting.

ABSTRACT

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.

Motivation & Objective

• To bridge the gap between theoretical security proofs of quantum key distribution and practical implementation by applying formal verification techniques.
• To evaluate the security of the BB84 protocol under realistic assumptions, particularly focusing on eavesdropping detection and information leakage.
• To demonstrate the effectiveness of automated model checking in analyzing quantum information systems where analytical solutions are complex or infeasible.
• To enable systematic exploration of protocol variations and implementation-level imperfections through a formal, extensible model.

Proposed method

• Employed the PRISM probabilistic model checker to formally verify the BB84 protocol using a state-based model of quantum state transmission and eavesdropping.
• Defined system behavior using probabilistic transition models that simulate Alice's preparation of qubits, Eve's intercept-resend attack, and Bob's measurement in random bases.
• Formulated security properties using PCTL (Probabilistic Computation Tree Logic), including the probability of detecting Eve and the likelihood of Eve measuring more than half the qubits correctly.
• Used nonlinear curve fitting (Levenberg–Marquardt) to model the exponential decay of eavesdropping success and detection probability as a function of qubit count.
• Enabled parameter variation in the model to explore effects of basis bias and other protocol modifications.
• Validated results against Mayers’ information-theoretic security proof by showing exponential convergence of security metrics with increasing qubit number.

Experimental results

Research questions

• RQ1 How does the probability of detecting an eavesdropper in the BB84 protocol scale with the number of transmitted qubits?
• RQ2 To what extent does the eavesdropper’s information gain about the key decrease as the number of qubits increases?
• RQ3 Can automated model checking tools like PRISM accurately reproduce known theoretical security properties of quantum key distribution?
• RQ4 How easily can the model be extended to analyze variations of BB84, such as biased basis selection or implementation-specific imperfections?
• RQ5 What is the quantitative relationship between the number of transmitted qubits and the exponential decay of eavesdropping success and detection probability?

Key findings

• The probability of detecting an eavesdropper in the BB84 protocol increases exponentially with the number of transmitted qubits, following the approximation $ P_{\text{det}}(n) \approx 1 - \exp[-0.134n] $.
• The probability that Eve measures more than half the qubits correctly decreases exponentially, with the approximation $ P_{>1/2}(n) \approx 0.909 \cdot \exp[-0.081n] $.
• The model confirms that eavesdropper information gain decreases exponentially with increasing qubit count $ n $, consistent with Mayers’ information-theoretic security claim.
• The results are robust within the tested range $ 5 \leq n \leq 30 $, showing strong exponential trends for both detection and information leakage.
• The model is extensible and can be adapted to analyze protocol variants, such as biased basis selection, by modifying the underlying PRISM model.
• The approach provides a practical, scalable, and automated method for verifying quantum cryptographic protocols, surpassing the limitations of purely analytical proofs in complex scenarios.