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.
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.
The Shor-Preskill proof of the security of the Bennett-Brassard 1984 (BB84) quantum key distribution protocol relies on the theoretical existence of good classical error-correcting codes with the ``dual-containing'' property. A practical implementation of the BB84 protocol thus requires explicit and efficiently decodable constructions of such codes, which are not known. On the other hand, modern coding theory abounds with non-dual-containing codes with excellent performance and efficient decoding algorithms. We show that the dual-containing constraint can be lifted at a small price: instead of a key distribution protocol, an efficiently implementable key expansion protocol is obtained, capable of increasing the size of a preshared key by a constant factor.
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.
• A quantum key distribution (QKD) protocol allows two parties Alice and Bob to establish secret key through one-way quantum communications and classical public communications. With the aid of an entanglement-assisted (EA) quantum error correcting (QEC) code, QKD can be made to expand a secret key, and the process is known as quantum key expansion (QKE). • Luo and Devetak[1] derived a QKE protocol by modifying the EA entanglement-distillation (EAED) protocol with the use of EA Calderbank-Shor-Steane (CSS) QEC codes. • Our initial goal is to search for QEC codes that guarantee high net key rates for QKE. The Original QKE Protocol • Let C1 and C2 be two classical [n, k1] and [n, k2] codes with parity-check matrices H1 and H2. and c = rank(M), ∃ nonsingu-• Let M = H1H T 2 lar matrices T1 and T2 s.t. • For i = 1, 2, let
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 describe the popular BB84 protocol and critically examine its security proof as presented by Shor and Preskill. The proof requires the use of quantum error correcting codes called the Calderbank-Shor-Steanne (CSS) quantum codes. These quantum codes are constructed in the quantum domain from two suitable classical linear codes, one used to correct for bit-flip errors and the other for phase-flip errors. Consequently, as a prelude to the security proof, the report reviews the essential properties of linear codes, especially the concept of cosets, before building the quantum codes that are utilized in the proof. The proof considers a security entanglement-based protocol, which is subsequently reduced to a "Prepare and Measure" protocol similar in structure to the BB84 protocol, thus establishing the security of the BB84 protocol. The proof, however, is not without assumptions, which are also enumerated. The treatment throughout is pedagogical, and this report, therefore, serves a useful tutorial for researchers, practitioners, and students, new to the field of quantum information science, in particular, quantum cryptography, as it develops the proof in a systematic manner, starting from the properties of linear codes, and then advancing to the quantum error correcting codes, which are critical to the understanding of the security proof.