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Physical Review A

Security against eavesdropping in quantum cryptography

Norbert Lütkenhaus

Published Jul 1, 1996• 130 citations
Publisher

Abstract

A sharp estimate is given for the amount of Shannon information and expected collision probability. This estimate is valid for all eavesdropping strategies described by a generalized measurement and restricted to the Hilbert space of the one-photon state. The optimal generalized measurement is explicitly given. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.97

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A sharp estimate is given for the amount of Shannon information and expected collision probability. This estimate is valid for all eavesdropping strategies described by a generalized measurement and restricted to the Hilbert space of the one-photon state. The optimal generalized measurement is explicitly given. \textcopyright{} 1996 The American Physical Society.
An attempt to eavesdrop on a quantum cryptographic channel reveals itself through errors it inevitably introduces into the transmission. We investigate the relationship between the induced error rate and the maximum amount of information the eavesdropper can extract, in both the two-state B92 [B92 refers to the work of C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992)] and the four-state BB84 [BB84 refers to the work of C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 1984), pp. 175--179] quantum cryptographic protocols. In each case, the optimal eavesdropping method that on average yields the most information for a given error rate is explicitly constructed. Analysis is limited to eavesdropping strategies where each bit of the quantum transmission is attacked individually and independently from other bits. Subject to this restriction, however, we believe that all attacks not forbidden by physical laws are covered. Unlike previous work, the eavesdropper's advantage is measured in terms of Renyi (rather than Shannon) information, and with respect only to bits received error-free by Bob (rather than all bits). This alters both the maximum extractable information and the optimal eavesdropping attack. The result can be used directly at the privacy amplification stage of the protocol to accomplish secure communication over a noisy channel.
When eavesdropping on a quantum channel, two distinct properties of quantum physics arise. First, if the quantum system is prepared in one of several non-orthogonal states, then no measurement can discriminate these states with certainty. Second, any measurement that acquires information about a system must necessarily disturb it. We investigate a simple and optimal eavesdropping scheme that minimizes the disturbance caused by a given amount of information gain, and show that an in principle demonstration of such a scheme could be performed using existing experimental techniques in single-photon quantum technologies.
In this article we deal with the security of the BB84 quantum cryptography protocol over noisy channels using generalized privacy amplification. For this we estimate the fraction of bits needed to be discarded during the privacy amplification step. This estimate is given for two scenarios, both of which assume the eavesdropper to access each of the signals independently and take error correction into account. One scenario does not allow a delay of the eavesdropper's measurement of a measurement probe until he receives additional classical information. In this scenario we achieve a sharp bound. The other scenario allows a measurement delay, so that the general attack of an eavesdropper on individual signals is covered. This bound is not sharp but allows a practical implementation of the protocol.
An attempt to eavesdrop a quantum cryptographic channel reveals itself through errors it inevitably introduces into the transmission. We investigate the relationship between the induced error rate and the maximum amount of information the eavesdropper can extract, both in the two-state B92 and the four-state BB84 quantum cryptographic protocols. In each case, the optimal eavesdropping method that on average yields the most information for a given error rate is explicitly constructed. Analysis is limited to eavesdropping strategies where each bit of the quantum transmission is attacked individually and independently form other bits. Subject to this restriction, however, we believe that all attacks not forbidden by physical laws are included. Unlike previous work, the eavesdropper's advantage is measured in terms of Renyi information, and with respect only to bits received error-free by Bob. This alters both the maximum extractable information and the optimal eavesdropping attack. The result can be used directly at the privacy amplification stage of the protocol to accomplish secure communication over a noisy channel.