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Noiseless attack on certain quantum key distribution protocols

Vinod N. Rao, R. Srikanth

Published Dec 9, 2020• 0 citations
OpenAlex

Abstract

The information-disturbance tradeoff lies at the heart of quantum key distribution (QKD), allowing a sender and a receiver to monitor eavesdropping in the communication channel by checking for errors in the transmission. Even so, here we show that for a class of QKD protocols based on the principle of interaction-free measurement, an eavesdropper Eve can obtain partial information noiselessly, i.e., in such a way that the disturbance produced does not show up during the error checking. The proposed attack is shown to be the optimal individual noiseless attack. An implication of the noiselessness is that even though error correction is not required, privacy amplification (PA) is necessary. For example, in the case of the protocol of [T.-G. Noh, PRL 103, 230501 (2009)], the compression factor of PA is 0.68 in the case of an individual, noiseless attack. For the noisy generalization of the above attack, where Eve introduces noise into the communication channel without adding noise to the eavesdropping channel, the security requirement entails a lower bound on the noise threshold of about 11%, assuming that the transmitted particles are single photons.

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The information-disturbance tradeoff lies at the heart of quantum key distribution (QKD), allowing a sender and a receiver to monitor eavesdropping in the communication channel by checking for errors in the transmission. Even so, here we show that for a class of QKD protocols based on the principle of interaction-free measurement, an eavesdropper Eve can obtain partial information noiselessly, i.e., in such a way that the disturbance produced does not show up during the error checking. The proposed attack is shown to be the optimal individual noiseless attack. An implication of the noiselessness is that even though error correction is not required, privacy amplification (PA) is necessary. For example, in the case of the protocol of [T.-G. Noh, PRL 103, 230501 (2009)], the compression factor of PA is 0.68 in the case of an individual, noiseless attack. For the noisy generalization of the above attack, where Eve introduces noise into the communication channel without adding noise to the eavesdropping channel, the security requirement entails a lower bound on the noise threshold of about 11%, assuming that the transmitted particles are single photons.
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.
Privacy amplification is a necessary step in all quantum key distribution protocols, and error correction is needed in each except when signals of many photons are used in the key communication in quantum noise approach. No security analysis of error correcting code information leak to the attacker has ever been provided, while an ad hoc formula is currently employed to account for such leak in the key generation rate. It is also commonly believed that privacy amplification allows the users to at least establish a short key of arbitrarily close to perfect security. In this paper we show how the lack of rigorous error correction analysis makes the otherwise valid privacy amplification results invalid, and that there exists a limit on how close to perfect a generated key can be obtained from privacy amplification. In addition, there is a necessary tradeoff between key rate and security, and the best theoretical values from current theories would not generate enough near-uniform key bits to cover the message authentication key cost in disturbance-information tradeoff protocols of the BB84 variety.
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.
We study eavesdropping in quantum key distribution with the six state protocol, when the signal states are mixed with white noise. This situation may arise either when Alice deliberately adds noise to the signal states before they leave her lab, or in a realistic scenario where Eve cannot replace the noisy quantum channel by a noiseless one. We find Eve's optimal mutual information with Alice, for individual attacks, as a function of the qubit error rate. Our result is that added quantum noise reduces Eve's mutual information more than Bob's.
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.
In conventional quantum key distribution (QKD) protocols, the information leak to an eavesdropper is estimated through the basic principle of quantum mechanics dictated in the original version of Heisenberg's uncertainty principle. The amount of leaked information on a shared sifted key is bounded from above essentially by using information-disturbance trade-off relations, based on the amount of signal disturbance measured via randomly sampled or inserted probe signals. Here we discuss an entirely different avenue toward the private communication, which does not rely on the information disturbance trade-off relations and hence does not require a monitoring of signal disturbance. The independence of the amount of privacy amplification from that of disturbance tends to give it a high tolerance on the channel noises. The lifting of the burden of precise statistical estimation of disturbance leads to a favorable finite-key-size effect. A protocol based on the novel principle can be implemented by only using photon detectors and classical optics tools: a laser, a phase modulator, and an interferometer. The protocol resembles the differential-phase-shift QKD protocol in that both share a simple binary phase shift keying on a coherent train of weak pulses from a laser. The difference lies in the use of a variable-delay interferometer in the new protocol, which randomly changes the combination of pulse pairs to be superposed. This extra randomness has turned out to be enough to upper-bound the information extracted by the eavesdropper, regardless of how they have disturbed the quantum signal.