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Journal of the Physical Society of Japan

Notes on a Continuous-Variable Quantum Key Distribution Scheme

Tsubasa Ichikawa, Takuya Hirano, Takuto Matsubara, Motoharu Ono, Ryo Namiki

Published Aug 10, 2017• 1 citation
Publisher

Abstract

We develop a physical model to describe the signal transmission for a continuous-variable quantum key distribution scheme and investigate its security against a couple of eavesdropping attacks assuming that the eavesdropper’s power is partly restricted owing to today’s technological limitations. We consider an eavesdropper performing quantum optical homodyne measurement on the signal obtained by a type of beamsplitting attack. We also consider the case in which the eavesdropper Eve is unable to access a quantum memory and she performs heterodyne measurement on her signal without performing a delayed measurement. Our formulation includes a model in which the receiver’s loss and noise are unaccessible by the eavesdropper. This setup enables us to investigate the condition that Eve uses a practical fiber differently from the usual beamsplitting attack where she can deploy a lossless transmission channel. The secret key rates are calculated in both the direct and reverse reconciliation scenarios.

DOI
10.7566/jpsj.86.094001

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We develop a physical model to describe the signal transmission for a continuous-variable quantum key distribution scheme and investigate its security against a couple of eavesdropping attacks assuming that the eavesdropper’s power is partly restricted owing to today’s technological limitations. We consider an eavesdropper performing quantum optical homodyne measurement on the signal obtained by a type of beamsplitting attack. We also consider the case in which the eavesdropper Eve is unable to access a quantum memory and she performs heterodyne measurement on her signal without performing a delayed measurement. Our formulation includes a model in which the receiver’s loss and noise are unaccessible by the eavesdropper. This setup enables us to investigate the condition that Eve uses a practical fiber differently from the usual beamsplitting attack where she can deploy a lossless transmission channel. The secret key rates are calculated in both the direct and reverse reconciliation scenarios.
We have developed a method to calculate a secret key rate of a continuous-variable quantum-key-distribution scheme using four coherent states and postselection for a general model of Gaussian attacks. We assume that the transmission line and detection process are described by a pair of Gaussian channels. In our analysis, while the loss and noise on the transmission line are induced by an eavesdropper, Eve, who can replace the transmission line with a lossless and noiseless optical fiber, she is assumed inaccessible to the detection process. By separating the transmission noise and detection noise, we can always extract a larger key compared with the case that all loss and noises are induced by an eavesdropper's interference. An asymptotic key rate against collective Gaussian attacks can be determined numerically for the given channels' parameters. The improvement of the key rates turns out to be more significant for the reverse-reconciliation scheme.
Research on optimal eavesdropping models under practical conditions will help to evaluate realistic risk when employing a quantum key distribution system for secure information transmission. Intuitively, fiber loss will lead to the optical energy leaking to the environment, rather than harvested by the eavesdropper, which also limits the eavesdropping ability while improving the quantum key distribution system performance in practical use. However, defining the optimal eavesdropping model in the presence of lossy fiber is difficult because the channel is beyond the control of legitimate partners and the leaked signal is undetectable. Here we investigate how the fiber loss influences the eavesdropping ability based on a teleportation-based collective attack model, which requires two distant stations and a shared entanglement source. We find that if the distributed entanglement is limited due to the practical loss, the optimal attack occurs when the two teleportation stations are merged to one and placed close to the transmitter site, which performs similar to the entangling-cloning attack but with a reduced wiretapping ratio. Assuming Eve uses the best available hollow-core fiber, the secret key rate in the practical environment can be 20--40% higher than that under ideal eavesdropping. While if the entanglement distillation technology is mature enough to provide high quality of distributed entanglement, the two teleportation stations should be distantly separated for better eavesdropping performance, where the eavesdropping can even approach the optimal collective attack. Under the current level of entanglement purification technology, the unavoidable fiber loss can still greatly limit the eavesdropping ability as well as enhance the secret key rate and transmission distance of the realistic system, which promotes the development of quantum key distribution systems in practical application scenarios.
We expand the recently discussed continuous-variable quantum key distribution scheme of Heid and Luetkenhaus (2006) to qudits with a lossy but noiseless quantum channel. Postselection methods are used. Secret key rates are calculated in the presence of loss for the direct and the reverse reconciliation method. In the case of reverse reconciliation the secret key rate increases with higher dimensions, in the case of direct reconciliation the result is reverse. Several ways of optimising the amplitude are discussed. In a next step the used states are generalised to squeezed states. Again secret key rates are calculated. The last part deals with the dual-homodyne detection which replaces the single-homodyne measurement. Most results are analytical and allow further treatments if the preparation symmetry stays unchanged.
In this paper we propose a continuous-variable subcarrier wave quantum key distribution scheme with homodyne detection and present its security analysis in the finite-size regime. We calculate a lower bound on the secret key rate for the system under the assumption that the quantum channel noise is negligible compared to detector dark counts, and an eavesdropper is restricted to collective attacks. Our calculation shows that the current realistic system implementation would allow distributing a secret keys over channels with losses up to 9 dB.
Quantum continuous variables (CV) have been explored as an alternative to qubits for quantum key distribution (QKD). More specifically, protocols using coherent states and homodyne or heterodyne measurements have been proposed and experimentally demonstrated (Grosshans et al., 2003). Relying on technologies allowing much higher rates than allowed by the single photon detectors used in qubit based QKD, those protocols are the only ones which could allow key rates in the GHz range in the foreseeable future. Furthermore, their security proofs have recently been extendend to generic attacks in the case of gaussian modulation. However, to our knowledge, all security analyses up to know have implicitely assumed Bob's measurement setup to use a homo- or heterodyne detection with an external classical phase reference, while the experimental realizations use a lightbeam for this role. This beam doesn't carry any information useful to the spy, and any attack on its phase could equally be carried on the signal beam. However, the intensity of this beam is used in experimental implementations to calibrate the quantum noise level. This calibration is then used to determine the channel gain, which is an important parameter of Eve's attack. We show that this allows Eve to implement an intercepte resend attack forbiding QKD as soon as the losses are above 3dB. We also give simple countermeasures to this new class of attacks.