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arXiv:quant-ph/0308048 · Physical Review A

Unconditional security of the Bennett 1992 quantum key-distribution protocol over a lossy and noisy channel

Kiyoshi Tamaki, Norbert Lütkenhaus

Published Mar 22, 2004• 76 citations
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Abstract

We show that the security proof of the Bennett 1992 protocol over loss-free channel given by Tamaki, Koashi, and Imoto [Phys. Rev. Lett. 90, 167904 (2003)] can be adapted to accommodate loss. We assumed that Bob's detectors discriminate between single-photon states on one hand and vacuum state or multiphoton states on the other hand.

DOI
10.1103/physreva.69.032316

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We show that the security proof of the Bennett 1992 protocol over loss-free channel given by Tamaki, Koashi, and Imoto [Phys. Rev. Lett. 90, 167904 (2003)] can be adapted to accommodate loss. We assumed that Bob's detectors discriminate between single-photon states on one hand and vacuum state or multiphoton states on the other hand.
Basic techniques to prove the unconditional security of quantum crypto graphy are described. They are applied to a quantum key distribution protocol proposed by Bennett and Brassard [1984]. The proof considers a practical variation on the protocol in which the channel is noisy and photos may be lost during the transmission. Each individual signal sent into the channel must contain a single photon or any two-dimensional system in the exact state described in the protocol. No restriction is imposed on the detector used at the receiving side of the channel, except that whether or not the received system is detected must be independent of the basis used to measure this system.
We prove the unconditional security of the original Bennett 1992 protocol with strong reference pulse. We show that we may place a projection onto suitably defined qubit spaces before the receiver, which makes the analysis as simple as qubit-based protocols. Unlike the single-photon-based qubits, the qubits identified in this scheme are almost surely detected by the receiver even after a lossy channel. This leads to the key generation rate that is proportional to the channel transmission rate for proper choices of experimental parameters.
In a deterministic quantum key distribution (DQKD) protocol with a two-way quantum channel, Bob sends a qubit to Alice who then encodes a key bit onto the qubit and sends it back to Bob. After measuring the returned qubit, Bob can obtain Alice's key bit immediately, without basis reconciliation. Since an eavesdropper may attack the qubits traveling on either the Bob-Alice channel or the Alice-Bob channel, the security analysis of DQKD protocol with a two-way quantum channel is complicated and its unconditional security has been controversial. This paper presents a security proof of a single-photon four-state DQKD protocol against general attacks.
We prove the unconditional security of a quantum key distribution (QKD) protocol on a noisy channel against the most general attack allowed by quantum physics. We use the fact that in a previous paper we have reduced the proof of the unconditionally security of this QKD protocol to a proof that a corresponding Quantum String Oblivious Transfer (String-QOT) protocol would be unconditionally secure against Bob if implemented on top of an unconditionally secure bit commitment schcmc. We prove a lemma that extends a security proof given by Yao for a (one bit) QOT protocol to this String-QOT protocol. This result and the reduction mentioned above implies the unconditional security of our QKD protocol despite our previous proof that unconditionally secure bit commitment schemes are impossible.