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Physical Review A

Simple proof of the unconditional security of the Bennett 1992 quantum key distribution protocol

Quan Zhang, Tang Chaojing

Published May 28, 2002• 20 citations

Abstract

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.

DOI: 10.1103/physreva.65.062301

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We prove the unconditional security of the standard six-state scheme for quantum key distribution (QKD). We demonstrate its unconditional security up to a bit error rate of 12.7 percents, by allowing only one-way classical communications in the error correction/privacy amplification procedure between Alice and Bob. This shows a clear advantage of the six-state scheme over another standard scheme---BB84, which has been proven to be secure up to only about 11 percents, if only one-way classical communications are allowed. Our proof technique is a generalization of that of Shor-Preskill's proof of security of BB84. We show that a advantage of the six-state scheme lies in the Alice and Bob's ability to establish rigorously from their test sample the non-trivial mutual information between the bit-flip and phase error patterns. A modified version of the degenerate quantum codes studied by DiVincenzo, Shor and Smolin is employed in our proof.

We devise a simple modification that essentially doubles the efficiency of the BB84 quantum key distribution scheme proposed by Bennett and Brassard. We also prove the security of our modified scheme against the most general eavesdropping attack that is allowed by the laws of physics. The first major ingredient of our scheme is the assignment of significantly different probabilities to the different polarization bases during both transmission and reception, thus reducing the fraction of discarded data. A second major ingredient of our scheme is a refined analysis of accepted data: We divide the accepted data into various subsets according to the basis employed and estimate an error rate for each subset *separately*. We then show that such a refined data analysis guarantees the security of our scheme against the most general eavesdropping strategy, thus generalizing Shor and Preskill's proof of security of BB84 to our new scheme. Up till now, most proposed proofs of security of single-particle type quantum key distribution schemes have relied heavily upon the fact that the bases are chosen uniformly, randomly and independently. Our proof removes this symmetry requirement.

Is the newly born quantum cryptography the ultimate solution for information security?A technique needs to be both theoretically strong and practically viable.But quantum cryptography comes to naught in the latter.Unlike many of the classical cryptosystems in use today, whose security often draws on unproven assumptions about the computational complexity of mathematical problems, the security of quantum cryptography is based on-and employs-the laws of physics.The term -unconditional security‖is used to emphasize the fact that it does not rely on the presumed, yet unproven hardness of somemathematical problem.In this Paper, we present the proof of the unconditional security of the BB84 protocol, as devised by Peter Shor and John Preskill [1].

Recently Boykin, Biham, Boyer, Mor, and Roychowdhury (BBBMR) proved the security of standard quantum key distribution against the most general attacks which can be performed on the channel [1]. The attack is performed by an eavesdropper who has unlimited computation abilities, and who uses the full power allowed by the rules of classical and quantum physics. The final key can then be used to transmit secure messages in a way that their security is also unaffected in the future. We explain here some of the subtleties related to the term “proving security”. We also present a simplified protocol, and we explain the role it plays in proving the security of the standard Bennett-Brassard protocol for quantum key distribution.

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.