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Physics Open

Security of quantum-key-distribution protocol by using the post-selection technique

Comfort Sekga, Mhlambululi Mafu

Published May 1, 2021• 13 citations
Publisher

Abstract

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.

DOI
10.1016/j.physo.2021.100075

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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.
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.
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.
Devices and protocols for information processing are often required to work for arbitrary inputs. For example, in channel coding theory, one demands that a coding scheme transmits any possible input state reliably over a given noisy channel. Similarly, in quantum cryptography, security of a protocol should hold independently of the inputs, even if they are chosen maliciously. In this short paper, we review the Post-Selection Technique introduced in (Phys. Rev. Lett. 102:020504, 2009). Its main purpose is to simplify the analysis of information processing schemes so that only one single input needs to be considered. If a scheme satisfies a desired criterion when acting on this particular input, then—under certain symmetry conditions—the same criterion is automatically met for arbitrary inputs. We illustrate the Post-Selection Technique at the example of quantum cryptography. Here, it can be used to show that security of a Quantum Key Distribution scheme against general attacks—somewhat surprisingly—follows from its security against one specific attack. This not only simplifies security proofs, but also has other remarkable implications, e.g., that no randomness is needed for privacy amplification.
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.