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arXiv:1008.2596 · New Journal of Physics

Finite-key security against coherent attacks in quantum key distribution

Lana Sheridan, Lê Phuc Thinh, Valerio Scarani

Published Dec 9, 2010• 81 citations
arXivHTMLPDF

Abstract

Christandl et al (2009 Phys. Rev. Lett. 102 020504) provide, in particular, the possibility of studying unconditional security in the finite-key regime for all discrete-variable protocols. We spell out this bound from their general formalism. Then, we apply it in the analysis of a recently proposed protocol (Laing et al 2010 Phys. Rev. A 82 012304). This protocol is meaningful when the alignment of Alice's and Bob's reference frames is not monitored and may vary with time. In this scenario, the notion of asymptotic key rate has hardly any operational meaning, because if one waits too long a time, the average correlations are smeared out and no security can be inferred. Therefore, finite-key analysis is necessary for finding the maximal achievable secret key rate and the corresponding optimal number of signals.

DOI
10.1088/1367-2630/12/12/123019

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Christandl et al (2009 Phys. Rev. Lett. 102 020504) provide, in particular, the possibility of studying unconditional security in the finite-key regime for all discrete-variable protocols. We spell out this bound from their general formalism. Then, we apply it in the analysis of a recently proposed protocol (Laing et al 2010 Phys. Rev. A 82 012304). This protocol is meaningful when the alignment of Alice's and Bob's reference frames is not monitored and may vary with time. In this scenario, the notion of asymptotic key rate has hardly any operational meaning, because if one waits too long a time, the average correlations are smeared out and no security can be inferred. Therefore, finite-key analysis is necessary for finding the maximal achievable secret key rate and the corresponding optimal number of signals.
We introduce a constructive method to calculate the achievable secret key rate for a generic class of quantum key distribution protocols, when only a finite number nnn of signals is given. Our approach is applicable to all scenarios in which the quantum state shared by Alice and Bob is known. In particular, we consider the six state protocol with symmetric eavesdropping attacks, and show that for a small number of signals, i.e., below $n\ensuremath{\sim}{10}^{4}$, the finite key rate differs significantly from the asymptotic value for $n\ensuremath{\rightarrow}\ensuremath{\infty}$. However, for larger nnn, a good approximation of the asymptotic value is found. We also study secret key rates for protocols using higher-dimensional quantum systems.
We derive a bound for the security of quantum key distribution with finite resources under one-way postprocessing, based on a definition of security that is composable and has an operational meaning. While our proof relies on the assumption of collective attacks, unconditional security follows immediately for standard protocols such as Bennett-Brassard 1984 and six-states protocol. For single-qubit implementations of such protocols, we find that the secret key rate becomes positive when at least N approximately 10(5) signals are exchanged and processed. For any other discrete-variable protocol, unconditional security can be obtained using the exponential de Finetti theorem, but the additional overhead leads to very pessimistic estimates.
The unconditional security of continuous-variable quantum key distribution is established for all schemes based on the estimation of the channel loss and excess noise. It is proved that, in the limit of large keys, Gaussian attacks are asymptotically optimal among the most general (coherent) attacks, where the transmission is tapped using arbitrary ancillas and stored in a quantum memory as a whole. Then, it is shown that the previously derived bounds on the achievable secret key rates against collective attacks remain asymptotically valid for arbitrary coherent attacks.
Continuous variable quantum key distribution with discrete modulation has the potential to provide information-theoretic security using widely available optical elements and existing telecom infrastructure. While their implementation is significantly simpler than that for protocols based on Gaussian modulation, proving their finite-size security against coherent attacks poses a challenge. In this work we prove finite-size security against coherent attacks for a discrete-modulated quantum key distribution protocol involving four coherent states and heterodyne detection. To do so, and contrary to most of the existing schemes, we first discretize all the continuous variables generated during the protocol. This allows us to use the entropy accumulation theorem, a tool that has previously been used in the setting of discrete variables, to construct the finite-size security proof. We then compute the corresponding finite-key rates through semi-definite programming and under a photon-number cutoff. Our analysis provides asymptotic rates in the range of 0.1 − 10 − 4 bits per round for distances up to hundred kilometres, while in the finite case and for realistic parameters, we get of the order of 10 Gbits of secret key after n ∼ 10 11 rounds and distances of few tens of kilometres.
Abstract Unlike traditional communication, quantum key distribution (QKD) can reach unconditional security and thus attracts intensive studies. Among all existing QKD protocols, round-robin-differential-phase-shift (RRDPS) protocol can be running without monitoring signal disturbance, which significantly simplifies its flow and improves its tolerance of error rate. Although several security proofs of RRDPS have been given, a tight finite-key analysis with a practical phase-randomized source is still missing. In this paper, we propose an improved security proof of RRDPS against the most general coherent attack based on the entropic uncertainty relation. What’s more, with the help of Azuma’s inequality, our proof can tackle finite-key effects primely. The proposed finite-key analysis keeps the advantages of phase randomization source and indicates experimentally acceptable numbers of pulses are sufficient to approach the asymptotical bound closely. The results shed light on practical QKD without monitoring signal disturbance.
Quantum key distribution (QKD) can be proved to be secure by laws of quantum mechanics. In this thesis, we review security proof methods in Renner's framework and discuss numerical methods to calculate asymptotic and finite key rates. These methods are highly versatile and applicable to general device-dependent QKD protocols. We also discuss analytical tools that extend the applicability of these numerical methods. We then present the asymptotic security proof against collective attacks for a variant of the twin-field QKD protocol, which can overcome the repeaterless secret-key capacity bound. Our variant reduces the sifting cost and uses non-phase-randomized coherent states as both signals and test states. We confirm the loss scaling of this protocol. Another important family of protocols that we investigate here are discrete-modulated continuous-variable QKD protocols. They are interesting due to their experimental simplicity and their great potential for massive deployment in the quantum-secured networks. Our security proof method can provide tight asymptotic key rates. We demonstrate that the postselection of data in combination with reverse reconciliation can improve the key rates. We analyze both untrusted and trusted detector noise scenarios. Our results in the trusted detector noise scenario show that we can thus cut out most of the effect of detector noise and obtain asymptotic key rates similar to those had we access to ideal detectors. Finally, we present several simple examples to illustrate our newly developed method for the numerical finite-key analysis against the most general attacks via the entropy accumulation theorem.