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arXiv:quant-ph/0212161 · Physical Review A

Security of the Bennett 1992 quantum-key distribution protocol against individual attack over a realistic channel

Kiyoshi Tamaki, Masato Koashi, Nobuyuki Imoto

Published Mar 26, 2003• 24 citations
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Abstract

The security of two-state quantum-key distribution against individual attack is estimated when the channel has losses and noises. We assume that Alice and Bob use two nonorthogonal single-photon polarization states. To make our analysis simple, we propose a modified $B92$ protocol in which Alice and Bob make use of inconclusive results, and Bob performs a kind of symmetrization of received states. Using this protocol, Alice and Bob can estimate Eve's information gain as a function of a few parameters that reflect the imperfections of devices, or Eve's disturbance. In some parameter regions, Eve's maximum information gain shows counterintuitive behavior, namely, it decreases as the amount of disturbances increases. For a small noise rate, Eve can extract perfect information in the case where the angle between Alice's two states is small or large, while she cannot extract perfect information for intermediate angles. We also estimate the secret key gain, which is the net growth of the secret key per pulse. We show the region where the modified $B92$ protocol over a realistic channel is secure against individual attack.

DOI
10.1103/physreva.67.032310

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Security analysis of the Bennett 1992 quantum key-distribution protocol over a realistic channel

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In this thesis, we analyze the security of the Bennett 1992 quantum key-distribution protocol (B92 protocol) over a realistic channel assuming that bit values are encoded in single photon polarization states. First, we study the security of the B92 protocol against individual attack. In the individual attack, eavesdropper (Eve) interacts a qubit emitted by the sender (Alice) with her probe system followed by a measurement on each probe. To make our analysis simple, we propose a modified B92 protocol. Using this protocol, Alice and the receiver (Bob) can estimate Eve's information gain as a function of a few parameters that reflect the imperfections of devices or Eve's disturbance. We find a counter-intuitive behavior of Eve's maximum information gain, i.e., it decreases as the amount of disturbances increases. We also estimate the secret key gain that is the net growth of the secret key per one pulse. We show the region where the modified B92 protocol over a realistic channel is secure against individual attack. Next, we study the unconditional security of the B92 protocol, which is the security against any attack. To prove the security, we first propose a protocol that is unconditionally secure and can be reduced to the B92 protocol. This protocol employs the entanglement distillation protocol (EDP) based on a filtering operation and the Calderbank-Shor-Steane (CSS) quantum error correcting codes. The bit errors and the phase errors, which have to be estimated for the EDP based on the CSS codes, are correlated after the filtering operation, and we can bound the amount of phase errors from the observed bit errors by an estimation method involving nonorthogonal measurements. The angle between the two states shows a trade-off between accuracy of the estimation and robustness to noises. We show a way to run the unconditionally secure B92.

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The security analysis of the Ekert 1991 (E91), Bennett 1992 (B92), six-state protocol, Scarani–Acín–Ribordy–Gisin 2004 (SARG04) quantum key distribution (QKD) protocols, and their variants have been studied in the presence of collective-rotation noise channels. However, besides the Bennett–Brassard 1984 (BB84) being the first proposed, extensively studied, and essential protocol, its security proof under collective-rotation noise is still missing. Thus, we aim to close this gap in the literature. Consequently, we investigate how collective-rotation noise channels affect the security of the BB84 protocol. Mainly, we study scenarios where the eavesdropper, Eve, conducts an intercept-resend attack on the transmitted photons sent via a quantum communication channel shared by Alice and Bob. Notably, we distinguish the impact of collective-rotation noise and that of the eavesdropper. To achieve this, we provide rigorous, yet straightforward numerical calculations. First, we derive a model for the collective-rotation noise for the BB84 protocol and parametrize the mutual information shared between Alice and Eve. This is followed by deriving the quantum bit error rate (QBER) for two intercept-resend attack scenarios. In particular, we demonstrate that, for small rotation angles, one can extract a secure secret key under a collective-rotation noise channel when there is no eavesdropping. We observe that noise induced by rotation of 0.35 radians of the prepared quantum state results in a QBER of 11%, which corresponds to the lower bound on the tolerable error rate for the BB84 QKD protocol against general attacks. Moreover, a rotational angle of 0.53 radians yields a 25% QBER, which corresponds to the error rate bound due to the intercept-resend attack. Finally, we conclude that the BB84 protocol is robust against intercept-resend attacks on collective-rotation noise channels when the rotation angle is varied arbitrarily within particular bounds.

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We discuss the unconditional security of a quantum key distribution protocol in which bit values are encoded in the phase of a weak coherent-state pulse relative to a strong reference pulse, which is essentially the one proposed by Bennett in 1992 (the B92 scheme). In the BB84 protocol with a perfect single photon source, the key rate decreases linearly with the transmission T of the channel. If we simply replace this source with a weak coherent-state pulse, the key rate drops more rapidly (as O(η2)) since the presence of multiple photons favors the eavesdropper, Eve. The B92 protocol, if modified to be encoded on the polarization of a single photon, also shows O(η2) dependence. This comes from Eve's option of sending the vacuum state, which is always registered as an inconclusive result and never causes bit errors. In the original B92 scheme with phase coding, Eve has no such option--replacing the weak pulse by the vacuum cannot avoid errors completely, thanks to the strong reference pulse. This should make the scheme stronger against the loss in the channel even coherent-state pulses from conventional lasers are used. We have obtained an unconditional security proof allowing the presence of small bit errors, and confirmed the above observation in a rigorous way. The key rate is found to drop linearly with the transmission η, and it is lower than the BB84 with a perfect single-photon source only by a constant.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jan 11, 2005•0 citations
The security of two-state quantum-key distribution against individual attack is estimated when the channel has losses and noises. We assume that Alice and Bob use two nonorthogonal single-photon polarization states. To make our analysis simple, we propose a modified B92B92B92 protocol in which Alice and Bob make use of inconclusive results, and Bob performs a kind of symmetrization of received states. Using this protocol, Alice and Bob can estimate Eve's information gain as a function of a few parameters that reflect the imperfections of devices, or Eve's disturbance. In some parameter regions, Eve's maximum information gain shows counterintuitive behavior, namely, it decreases as the amount of disturbances increases. For a small noise rate, Eve can extract perfect information in the case where the angle between Alice's two states is small or large, while she cannot extract perfect information for intermediate angles. We also estimate the secret key gain, which is the net growth of the secret key per pulse. We show the region where the modified B92B92B92 protocol over a realistic channel is secure against individual attack.
In this thesis, we analyze the security of the Bennett 1992 quantum key-distribution protocol (B92 protocol) over a realistic channel assuming that bit values are encoded in single photon polarization states. First, we study the security of the B92 protocol against individual attack. In the individual attack, eavesdropper (Eve) interacts a qubit emitted by the sender (Alice) with her probe system followed by a measurement on each probe. To make our analysis simple, we propose a modified B92 protocol. Using this protocol, Alice and the receiver (Bob) can estimate Eve's information gain as a function of a few parameters that reflect the imperfections of devices or Eve's disturbance. We find a counter-intuitive behavior of Eve's maximum information gain, i.e., it decreases as the amount of disturbances increases. We also estimate the secret key gain that is the net growth of the secret key per one pulse. We show the region where the modified B92 protocol over a realistic channel is secure against individual attack. Next, we study the unconditional security of the B92 protocol, which is the security against any attack. To prove the security, we first propose a protocol that is unconditionally secure and can be reduced to the B92 protocol. This protocol employs the entanglement distillation protocol (EDP) based on a filtering operation and the Calderbank-Shor-Steane (CSS) quantum error correcting codes. The bit errors and the phase errors, which have to be estimated for the EDP based on the CSS codes, are correlated after the filtering operation, and we can bound the amount of phase errors from the observed bit errors by an estimation method involving nonorthogonal measurements. The angle between the two states shows a trade-off between accuracy of the estimation and robustness to noises. We show a way to run the unconditionally secure B92.
We suggest here a two-point eavesdropping strategy to two nonorthogonal states protocol of quantum key distribution over a fiber-optic channel. Suppose that the single-photon sources and detectors of Alice, Bob and Eves are ideal ones, the two-point attack can break the two nonorthogonal states protocol if the distance between Alice and Bob is sufficiently long. For single-photon type B92 protocol, the upper bound of distance is about 30 kilometers. It is interesting that Bennett's original multi-photon protocol is secure against both two-point attack and beam splitting attack, even if the protocol is realized with weak pulse source.
Abstract This work presents a security analysis of the semi-quantum key distribution protocol when the transmitted states are modeled as a depolarizing channel i.e. a mixture of the ideal state and the maximally mixed state. This scenario can occur either by the deliberate addition of noise by Alice to the states before their transmission, or in a practical situation where Eve cannot substitute the noisy quantum channel with an ideal one. We derive the maximum amount of information that Eve can obtain from Alice in the case of individual attacks, for protocols employing two and three mutually unbiased bases. The findings show that depolarization decreases Eve’s information gain compared to that of Bob.
We suggest here a two-point eavesdropping strategy aimed at a two nonorthogonal states protocol of quantum key distribution over a fiber-optic channel. When the single-photon sources and detectors of Alice, Bob and the two Eve are all ideal, the two-point attack can break the two nonorthogonal states protocol if the distance between Alice and Bob is longer than 30 km. However, Bennett's original multi-photon protocol is secure against both two-point and beam splitting attacks, though the protocol is realized with a weak pulsed source.
The security analysis of the Ekert 1991 (E91), Bennett 1992 (B92), six-state protocol, Scarani–Acín–Ribordy–Gisin 2004 (SARG04) quantum key distribution (QKD) protocols, and their variants have been studied in the presence of collective-rotation noise channels. However, besides the Bennett–Brassard 1984 (BB84) being the first proposed, extensively studied, and essential protocol, its security proof under collective-rotation noise is still missing. Thus, we aim to close this gap in the literature. Consequently, we investigate how collective-rotation noise channels affect the security of the BB84 protocol. Mainly, we study scenarios where the eavesdropper, Eve, conducts an intercept-resend attack on the transmitted photons sent via a quantum communication channel shared by Alice and Bob. Notably, we distinguish the impact of collective-rotation noise and that of the eavesdropper. To achieve this, we provide rigorous, yet straightforward numerical calculations. First, we derive a model for the collective-rotation noise for the BB84 protocol and parametrize the mutual information shared between Alice and Eve. This is followed by deriving the quantum bit error rate (QBER) for two intercept-resend attack scenarios. In particular, we demonstrate that, for small rotation angles, one can extract a secure secret key under a collective-rotation noise channel when there is no eavesdropping. We observe that noise induced by rotation of 0.35 radians of the prepared quantum state results in a QBER of 11%, which corresponds to the lower bound on the tolerable error rate for the BB84 QKD protocol against general attacks. Moreover, a rotational angle of 0.53 radians yields a 25% QBER, which corresponds to the error rate bound due to the intercept-resend attack. Finally, we conclude that the BB84 protocol is robust against intercept-resend attacks on collective-rotation noise channels when the rotation angle is varied arbitrarily within particular bounds.
We discuss the unconditional security of a quantum key distribution protocol in which bit values are encoded in the phase of a weak coherent-state pulse relative to a strong reference pulse, which is essentially the one proposed by Bennett in 1992 (the B92 scheme). In the BB84 protocol with a perfect single photon source, the key rate decreases linearly with the transmission T of the channel. If we simply replace this source with a weak coherent-state pulse, the key rate drops more rapidly (as O(η2)) since the presence of multiple photons favors the eavesdropper, Eve. The B92 protocol, if modified to be encoded on the polarization of a single photon, also shows O(η2) dependence. This comes from Eve's option of sending the vacuum state, which is always registered as an inconclusive result and never causes bit errors. In the original B92 scheme with phase coding, Eve has no such option--replacing the weak pulse by the vacuum cannot avoid errors completely, thanks to the strong reference pulse. This should make the scheme stronger against the loss in the channel even coherent-state pulses from conventional lasers are used. We have obtained an unconditional security proof allowing the presence of small bit errors, and confirmed the above observation in a rigorous way. The key rate is found to drop linearly with the transmission η, and it is lower than the BB84 with a perfect single-photon source only by a constant.