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arXiv:quant-ph/0110079 · Physical Review A

Shor-Preskill-type security proof for concatenated Bennett-Brassard 1984 quantum-key-distribution protocol

Won-Young Hwang, Keiji Matsumoto, Hiroshi Imai, Jaewan Kim, Hai-Woong Lee

Published Feb 10, 2003• 2 citations
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Abstract

We discuss a long code problem in the Bennett-Brassard 1984 (BB84) quantum-key-distribution protocol and describe how it can be overcome by concatenation of the protocol. Observing that concatenated modified Lo-Chau protocol finally reduces to the concatenated BB84 protocol, we give the unconditional security of the concatenated BB84 protocol.

DOI
10.1103/physreva.67.024302

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We discuss a long code problem in the Bennett-Brassard 1984 (BB84) quantum-key-distribution protocol and describe how it can be overcome by concatenation of the protocol. Observing that concatenated modified Lo-Chau protocol finally reduces to the concatenated BB84 protocol, we give the unconditional security of the concatenated BB84 protocol.
It is generally accepted that quantum key distribution (QKD) could supply legitimate users with unconditional security during their communication. Quite a lot of satisfactory efforts have been achieved on experimentations with quantum cryptography. However, when the eavesdropper has extra-powerful computational ability, has access to a quantum computer, for example, and can carry into execution any eavesdropping measurement that is allowed by the laws of physics, the security against such attacks has not been widely studied and rigorously proved for most QKD protocols. Quite recently, Shor and Preskill proved concisely the unconditional security of the Bennett-Brassard 1984 (BB84) protocol. Their method is highly valued for its clarity of concept and concision of form. In order to take advantage of the Shor-Preskill technique in their proof of the unconditional security of the BB84 QKD protocol, we introduced in this paper a transformation that can translate the Bennett 1992 (B92) protocol into the BB84 protocol. By proving that the transformation leaks no more information to the eavesdropper, we proved the unconditional security of the B92 protocol. We also settled the problem proposed by Lo about how to prove the unconditional security of the B92 protocol with the Shor-Preskill method.
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.
We present an approach to the unconditional security of quantum key distribution protocols based on the uncertainty principle. The approach applies to every case that has been treated via the argument by Shor and Preskill, and relieve them from the constraints of finding quantum error correcting codes. It can also treat the cases with uncharacterized apparatuses. We derive a secure key rate for the Bennett-Brassard-1984 protocol with an arbitrary source characterized only by a single parameter representing the basis dependence.
Quantum key distribution (QKD) protocols make it possible for two parties to generate a secret shared key. One of the most important QKD protocols, BB84, was suggested by Bennett and Brassard in 1984. Various proofs of unconditional security for BB84 have been suggested, but the first security proofs were not composable. Here we improve a security proof of BB84 given by [Biham, Boyer, Boykin, Mor, and Roychowdhury, J. Cryptology, 19(4):381-439, 2006] to be composable and match the state-of-the-art results for BB84, and we extend it to prove unconditional security of several variants of the BB84 protocol. Our composable security proof for BB84 and its variants is mostly self-contained, algebraic, and relatively simple, and it gives tight finite-key bounds.
We give a Shor-Preskill-type security proof to quantum key distribution without public announcement of bases [W.Y. Hwang et al., Phys. Lett. A 244, 489 (1998)]. First, we modify the Lo-Chau protocol once more so that it finally reduces to the quantum key distribution without public announcement of bases. Then we show how we can estimate the error rate in the code bits based on that in the checked bits in the proposed protocol, which is the central point of the proof. We discuss the problem of imperfect sources and that of large deviation in the error rate distributions. We discuss when the bases sequence must be discarded.