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arXiv:quant-ph/0412035 · Physical Review A

Unconditionally secure key distillation from multiphotons

Kiyoshi Tamaki, Hoi‐Kwong Lo

Published Jan 17, 2006• 73 citations
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Abstract

In this paper, we prove that the unconditionally secure key can be surprisingly extracted from multiphoton emission part in the photon polarization-based quantum key distribution. One example is shown by explicitly proving that one can indeed generate an unconditionally secure key from Alice's two-photon emission part proposed by Scarani [ et al. Phys. Rev. Lett. 92, 057901 (2004)]. Which is called the Scarani-Acin-Ribordy-Gisin (SARG04) protocol. This protocol uses the same four states as in Bennett-Brassard 1984 (BB84) and differs only in the classical postprocessing protocol. It is, thus, interesting to see how the classical postprocessing of quantum key distribution might qualitatively change its security. We also show that one can generate an unconditionally secure key from the single to the four-photon part in a generalized SARG04 protocol that uses six states. Finally, we also compare the bit error rate threshold of these protocols with the one in the BB84 protocol and the original six-state protocol assuming a depolarizing channel.

DOI
10.1103/physreva.73.010302

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In this paper, we prove that the unconditionally secure key can be surprisingly extracted from multiphoton emission part in the photon polarization-based quantum key distribution. One example is shown by explicitly proving that one can indeed generate an unconditionally secure key from Alice's two-photon emission part proposed by Scarani [ et al. Phys. Rev. Lett. 92, 057901 (2004)]. Which is called the Scarani-Acin-Ribordy-Gisin (SARG04) protocol. This protocol uses the same four states as in Bennett-Brassard 1984 (BB84) and differs only in the classical postprocessing protocol. It is, thus, interesting to see how the classical postprocessing of quantum key distribution might qualitatively change its security. We also show that one can generate an unconditionally secure key from the single to the four-photon part in a generalized SARG04 protocol that uses six states. Finally, we also compare the bit error rate threshold of these protocols with the one in the BB84 protocol and the original six-state protocol assuming a depolarizing channel.
In this presentation, we show some counter-examples to a naive belief that the security of QKD is based on no-cloning theorem. One example is shown by explicitly proving that one can indeed generate an unconditionally secure key from Alice's two-photon emission part in "SARG04 protocol" proposed by V. Scarani et al, in Phys. Rev. Lett. 92, 057901 (2004). This protocol differs from BB84 only in the classical communication. It is, thus, interesting to see how only the classical communication of QKD protocol might qualitatively change its security. We also show that one can generate an unconditionally secure key from the single to the four-photon part in a generalized SARG04 that uses six states. Finally, we also compare the bit error rate threshold of these protocols with the one in BB84 and the original six-state protocol assuming a depolarizing channel
Most qubit-based quantum key distribution (QKD) protocols extract the secure key merely from single-photon component of the attenuated lasers. However, with the Scarani-Acin-Ribordy-Gisin 2004 (SARG04) QKD protocol, the unconditionally secure key can be extracted from the two-photon component by modifying the classical post-processing procedure in the BB84 protocol. Employing the merits of SARG04 QKD protocol and six-state preparation, one can extract secure key from the components of single photon up to four photons. In this paper, we provide the exact relations between the secure key rate and the bit error rate in a six-state SARG04 protocol with single-photon, two-photon, three-photon, and four-photon sources. By restricting the mutual information between the phase error and bit error, we obtain a higher secure bit error rate threshold of the multiphoton components than previous works. Besides, we compare the performances of the six-state SARG04 with other prepare-and-measure QKD protocols using decoy states.
We compare the performance of Bennett-Brassard 1984 (BB84) and Scarani-Acin-Ribordy-Gisin 2004 (SARG04) protocols, the latter of which was proposed by V. Scarani et al. [Phys. Rev. Lett. 92, 057901 (2004)]. Specifically, in this paper, we investigate the SARG04 protocol with two-way classical communications and the SARG04 protocol with decoy states. In the first part of the paper, we show that the SARG04 scheme with two-way communications can tolerate a higher bit error rate (19.4% for a one-photon source and 6.56% for a two-photon source) than the SARG04 one with one-way communications (10.95% for a one-photon source and 2.71% for a two-photon source). Also, the upper bounds on the bit error rate for the SARG04 protocol with two-way communications are computed in a closed form by considering an individual attack based on a general measurement. In the second part of the paper, we propose employing the idea of decoy states in the SARG04 scheme to obtain unconditional security even when realistic devices are used. We compare the performance of the SARG04 protocol with decoy states and the BB84 one with decoy states. We find that the optimal mean-photon number for the SARG04 scheme is higher than that of the BB84 one when the bit error rate is small. Also, we observe that the SARG04 protocol does not achieve a longer secure distance and a higher key generation rate than the BB84 one, assuming a typical experimental parameter set.
We prove the unconditional security of quantum key distribution protocols using attenuated laser pulses with M different linear polarizations. When M=4, the proof covers the so-called SARG04 protocol [V.~Scarani et al., Phys. Rev.\ Lett. {\bf 92}, 057901 (2004)], which uses exactly the same quantum communication as the Bennett-Brassard 1984 protocol. For a channel with transmission ηηη, we show that the key rate in SARG04 scales as O(η3/2)O(η^{3/2})O(η3/2). When we increase the number of states to M=2k-1 or 2k, the key rate scaling improves as O(η(k+1)/k)O(η^{(k+1)/k})O(η(k+1)/k).
The post-selection technique was proposed by Christandl, König, and Renner [Phys. Rev. Lett. 102, 020504 (2009)] to simplify the security of quantum-key-distribution schemes. This work applies this technique to derive the security bounds for the six-state Scarani-Acin-Rigbordy-Gisin 2004 quantum key distribution protocol. This protocol can extract secure keys from a source emitting multi-photons due to its robustness from photon number splitting attacks, making it a good candidate for practical implementations. We also compare the security bounds for this protocol under collective attacks to the security bounds obtained after applying the post-selection technique when using finite resources. Our results demonstrate that the bounds for optimal attack are close to the bound for collective attack for many signals. Notably, the six-state SARG04 protocol proves to be more robust to the PNS attacks when compared to the original four-state SARG04 protocol. This demonstrates the power of the post-selection technique in deriving the security bounds for the six-state protocol when finite resources are used.
The first quantum cryptography protocol, proposed by Bennett and Brassard in 1984 (BB84), has been widely studied in recent years. This protocol uses four states (more precisely, two complementary bases) for the encoding of the classical bit. Recently, it has been noticed that by using the same four states, but a different encoding of information, one can define a protocol which is more robust in practical implementations, specifically when attenuated laser pulses are used instead of single-photon sources [V. Scarani et al., Phys. Rev. Lett. 92, 057901 (2004), referred to as the SARG04 protocol]. We present a detailed study of SARG04 in two different regimes. In the first part, we consider an implementation with a single-photon source: we derive bounds on the error rate QQQ for security against all possible attacks by the eavesdropper. The lower and the upper bound obtained for SARG04 ($Q\ensuremath{\lesssim}10.95%$ and $Q\ensuremath{\gtrsim}14.9%$, respectively) are close to those obtained for BB84 ($Q\ensuremath{\lesssim}12.4%$ and $Q\ensuremath{\gtrsim}14.6%$, respectively). In the second part, we consider a realistic source consisting of an attenuated laser and improve on previous analysis by allowing Alice to optimize the mean number of photons as a function of the distance. The SARG04 protocol is found to perform better than BB84, both in secret-key rate and in maximal achievable distance, for a wide class of Eve's attacks.