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arXiv:quant-ph/0308029 · Journal of Physics A Mathematical and General

Reliability of Calderbank–Shor–Steane codes and security of quantum key distribution

Mitsuru Hamada

Published Aug 12, 2004• 42 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/37/34/009

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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.
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.
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.
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.
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.
The secure prepare-and-measure BB84 quantum key distribution (QKD) protocol implicit in the Shor-Preskill security proof is based on Calderbank-Shor-Steane (CSS) codes. The latter rely on the existence of pairs of good codes with mutually orthogonal parity check matices, which are hard to construct. Here we show that the orthogonality requirement can be lifted at a small price: instead of a QKD protocol we obtain a quantum key expansion (QKE) protocol, capable of increasing the size of a pre-shared key by a constant factor. Our construction is based on the recently discovered entanglement-assisted quantum codes of Brun, Devetak and Hsieh.
In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parametersQ1=[[N,Ω(N),Ω(N)]]\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]Q1​=[[N,Ω(N​),Ω(N​)]], andQ2=[[N,Ω(N/log⁡N),Ω(N/log⁡N)/Ω(log⁡N)]]\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]Q2​=[[N,Ω(N/logN),Ω(N/logN)/Ω(logN)]]. We show thatQ1\mathscr {Q}_{1}Q1​can be encoded very efficiently by circuits of sizeO(N)O(N)O(N)and depthO(N)O(\sqrt {N})O(N​). For an input error syndrome,Q1\mathscr {Q}_{1}Q1​can correct any adversarial error of weight up to half the minimum distance bound inO(N)O(N)O(N)time.Q1\mathscr {Q}_{1}Q1​can also be decoded in parallel inO(N)O(\sqrt {N})O(N​)time by usingO(N)O(\sqrt {N})O(N​)classical processors. For an input error syndrome, we proved thatQ2\mathscr {Q}_{2}Q2​can correct a linear number ofX{X}X-errors with high probability and an almost linear number ofZ{Z}Z-errors inO(N)O(N)O(N)time. Moreover,Q2\mathscr {Q}_{2}Q2​can be decoded in parallel inO(log⁡(N))O(\log (N))O(log(N))time by usingO(N)O(N)O(N)classical processors.