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International Journal of Quantum Information

The security analysis of the BB84 protocol in the case of Calderbank–Shor–Steane code leakage

Ming Fang, Yaping Li, Li Fei

Published Jan 6, 2022• 0 citations

Abstract

Quantum key distribution (QKD) allows authenticated parties to share secure keys. Its security comes from quantum physics rather than computational complexity. The previous work has been able to demonstrate the security of the BB84 protocol based on the uncertainty principle, entanglement purification and information theory. In the security proof method based on entanglement purification, it is assumed that the information of Calderbank–Shor–Steane (CSS) error correction code cannot be leaked, otherwise, it is insecure. However, there is no quantitative analysis of the relationship between the parameter of CSS code and the amount of information leaked. In the attack and defense strategy of the actual quantum key distribution system, especially in the application of the device that is easy to lose or out of control, it is necessary to assess the impact of the parameter leakage. In this paper, we derive the relationship between the leaked parameter of CSS code and the amount of the final key leakage based on the BB84 protocol. Based on this formula, we simulated the impact of different CSS code parameter leaks on the final key amount. Through the analysis of simulation results, the security of the BB84 protocol is inversely proportional to the value of [Formula: see text] and [Formula: see text] in the case of the CSS code leak.

DOI: 10.1142/s0219749921500428

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This paper describes the security of quantum key distribution (QKD) to share a random secret string of digits between two parties based on the principle of quantum mechanics. A Colderbank-Shor-Steane quantum code had been implicitly used in the BB84 protocol, which makes fidelity of this code to unity and the mutual information between the shared key and the data obtained approaches zero. In the BB84 protocol, the sender transmits digits from which encoded into an orthogonal basis or the conjugate basis. These CSS codes achieve a higher rate called as Shannon rate.

After Mayers (1996, 2001) gave a proof of the security of the Bennett-Brassard 1984 (BB84) quantum key distribution protocol, Shor and Preskill (2000) made a remarkable observation that a Calderbank-Shor-Steane (CSS) code had been implicitly used in the BB84 protocol, and suggested its security could be proven by bounding the fidelity, say F(n), of the incorporated CSS code of length n in the form 1-F(n) <= exp[-n E+o(n)] for some positive number E. This work presents such a number E=E(R) as a function of the rate of a code R, and a threshold R' such that E(R)>0 whenever R < R', which is larger than the achievable rate based on the Gilbert-Varshamov bound that is essentially due to Shor and Preskill (2000). The codes in the present work are robust against fluctuations of channel parameters, which fact is needed to establish the security rigorously and was not proved for rates above the Gilbert-Varshamov rate before in the literature. As a byproduct, the security of a modified BB84 protocol against any joint (coherent) attacks is proved quantitatively.

Quantum key distribution (QKD) enables secure underwater communications essential for maritime infrastructure. Underwater optical channels introduce substantial photon loss (erasures) and ambient noise that degrade QKD performance. This paper investigates whether two four-qubit Calderbank-Shor-Steane (CSS) quantum error correction codes (QECC) mitigate these impairments in vertical underwater communication BB84 QKD protocol. After developing a comprehensive stochastic channel model incorporating photon loss, geometric spreading, and solar noise, we assess the viability of QECC through the analytical study of the quantum bit error rate (QBER) and the secure key rate (SKR) with and without security depending on the signal-to-noise ratio (SNR) validated against Monte Carlo simulations. The standard four-qubit CSS code achieves a 3 dB SNR gain at the QBER 11% security threshold; the discard code variant achieves 4.5 dB. However, QECC includes an encoding overhead that reduces the SKR. We demonstrate a crucial relationship between the SKR and the probability of arrival of the sent photon. Analysis shows that QECC is beneficial exclusively in marginal SNR regimes; at high SNR, raw BB84 dominates. For an ocean Type III Jerlov water scenario with sunlight coming from the sun located at the top of the atmosphere, we identify operational depth-range windows where QECC enables communication otherwise infeasible. This analysis establishes that error correction deployment must be scenario-dependent: extend operational range at the cost of throughput when SNR is marginal, or prioritize key generation rates at high SNR.

We prove that the 1984 protocol of Bennett and Brassard (BB84) for quantum key distribution is secure. We first give a key distribution protocol based on entanglement purification, which can be proven secure using methods from Lo and Chau's proof of security for a similar protocol. We then show that the security of this protocol implies the security of BB84. The entanglement purification based protocol uses Calderbank-Shor-Steane codes, and properties of these codes are used to remove the use of quantum computation from the Lo-Chau protocol.

The BB84 quantum key distribution (QKD) protocol enables two authenticated parties to generate a secret key over an insecure quantum channel. Using a standardized security definition, we prove that BB84 is secure and include explicit bounds on its security. Furthermore, our use of quantum circuit diagrams simplify the Shor-Preskill proof. Namely, we can reduce the Modified Lo-Chau QKD to a practical version of BB84 using the observation from Shor and Preskill that one may ignore a correctable number of phase errors, and the fact that computational basis measurements commute with controls of CNOT operations. The first four chapters provide the required background material on quantum computing, information theory, cryptography, coding theory, and quantum error correcting codes. Chapter 5 presents protocols for entanglement purification. Chapter 6 reduces an entanglement purification protocol to the Modified Lo-Chau QKD, and proves that it is secure. Finally, a reduction from the Modified Lo-Chau QKD to BB84 establishes the security of the latter.

In the quantum key distribution (QKD) process, the presence of noise in the quantum channel reduces the efficiency of QKD with low transmission reliability. Quantum error correcting codes (QECCs) can weaken the impact of noise on the final result due to its flexibility in code length and rate selection. In this paper, QECCs is used to overcome the adverse effects of quantum channel noise, and a QKD protocol based on quantum CSS error correcting codes is proposed. Firstly, we construct the corresponding quantum CSS codes based on classical linear code, then modify and simplify some steps of BB84 protocol by using the characteristics of error correcting codes to design the new protocol. Through theoretical deduction, our QKD protocol based on quantum CSS codes improves the security of key transmission, and the confidentiality of transmission in the noisy quantum channel is also enhanced.