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arXiv:0708.0709 · Physical Review Letters

Quantum Cryptography with Finite Resources: Unconditional Security Bound for Discrete-Variable Protocols with One-Way Postprocessing

Valerio Scarani, Renato Renner

Published May 22, 2008• 357 citations
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Abstract

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.

DOI
10.1103/physrevlett.100.200501

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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.
We present a technique for proving the security of quantum-key-distribution (QKD) protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel.
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 investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret-key rate of those QKD protocols involving only entropies of two-qubit density operators. As an illustration of our results, we determine new bounds for the Bennett-Brassard 1984, the 6-state, and the Bennett 1992 protocols. We show that in all these cases the first classical processing that the legitimate partners should apply consists in adding noise.
We give the first composable security proof for continuous-variable quantum key distribution with coherent states against collective attacks. Crucially, in the limit of large blocks the secret key rate converges to the usual value computed from the Holevo bound. Combining our proof with either the de Finetti theorem or the postselection technique then shows the security of the protocol against general attacks, thereby confirming the long-standing conjecture that Gaussian attacks are optimal asymptotically in the composable security framework. We expect that our parameter estimation procedure, which does not rely on any assumption about the quantum state being measured, will find applications elsewhere, for instance, for the reliable quantification of continuous-variable entanglement in finite-size settings.
We consider continuous-variable quantum key distribution with discrete-alphabet encodings.In particular, we study protocols where information is encoded in the phase of displaced coherent (or thermal) states, even though the results can be directly extended to any protocol based on finite constellations of displaced Gaussian states.In this setting, we study the composable security in the finite-size regime assuming the realistic but restrictive hypothesis of collective Gaussian attacks.Under this assumption, we can efficiently estimate the parameters of the channel via maximum likelihood estimators and bound the corresponding error in the final secret key rate.