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arXiv:1302.4139 · New Journal of Physics

Security analysis of the decoy method with the Bennett–Brassard 1984 protocol for finite key lengths

Masahito Hayashi, Ryota Nakayama

Published Jun 5, 2014• 69 citations
arXivHTMLPDF

Abstract

This paper provides a formula for the sacrifice bit-length for privacy amplification with the Bennett–Brassard 1984 protocol for finite key lengths, when we employ the decoy method. Using the formula, we can guarantee the security parameter for a realizable quantum key distribution system. The key generation rates with finite key lengths are numerically evaluated. The proposed method improves the existing key generation rate even in the asymptotic setting.

DOI
10.1088/1367-2630/16/6/063009

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This paper provides a formula for the sacrifice bit-length for privacy amplification with the Bennett–Brassard 1984 protocol for finite key lengths, when we employ the decoy method. Using the formula, we can guarantee the security parameter for a realizable quantum key distribution system. The key generation rates with finite key lengths are numerically evaluated. The proposed method improves the existing key generation rate even in the asymptotic setting.
We present a tight security analysis of the Bennett–Brassard 1984 protocol taking into account the finite-size effect of key distillation and achieving unconditional security. We begin by presenting a concise analysis utilizing the normal approximation of the hypergeometric function. Next we show that a similar tight bound can also be obtained by a rigorous argument without relying on any approximation. In particular, for the convenience of experimentalists who wish to evaluate the security of their quantum key distribution systems, we also give the explicit procedures of our key distillation and show how to calculate the secret key rate and the security parameter from a given set of experimental parameters. In addition to the exact values of key rates and security parameters, we also describe how to obtain their rough estimates using the normal approximation.
The decoy-state Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol is widely regarded as the de facto standard for practical implementations. On the receiver side, passive basis choice is attractive because it significantly reduces the need for random number generators and eliminates the need for optical modulators. Despite these advantages, a finite-key analytical security proof for the decoy-state BB84 protocol, where the basis is chosen passively with a biased probability, has been lacking. In this work, we present a simple analytical finite-key security proof for this setting, yielding a closed-form secret-key rate formula that can be directly evaluated using experimentally accessible parameters. Numerical simulations show that the key rates of passiveand active-measurement implementations are nearly identical, indicating that passive measurement does not compromise key-generation efficiency in practical QKD systems.
In this work we present a security analysis for quantum key distribution, establishing a rigorous tradeoff between various protocol and security parameters for a class of entanglement-based and prepare-and-measure protocols. The goal of this paper is twofold: 1) to review and clarify the stateof-the-art security analysis based on entropic uncertainty relations, and 2) to provide an accessible resource for researchers interested in a security analysis of quantum cryptographic protocols that takes into account finite resource effects. For this purpose we collect and clarify several arguments spread in the literature on the subject with the goal of making this treatment largely self-contained. More precisely, we focus on a class of prepare-and-measure protocols based on the Bennett-Brassard (BB84) protocol as well as a class of entanglement-based protocols similar to the Bennett-Brassard-Mermin (BBM92) protocol. We carefully formalize the different steps in these protocols, including randomization, measurement, parameter estimation, error correction and privacy amplification, allowing us to be mathematically precise throughout the security analysis. We start from an operational definition of what it means for a quantum key distribution protocol to be secure and derive simple conditions that serve as sufficient condition for secrecy and correctness. We then derive and eventually discuss tradeoff relations between the block length of the classical computation, the noise tolerance, the secret key length and the security parameters for our protocols. Our results significantly improve upon previously reported tradeoffs.
Decoy state method quantum key distribution (QKD) is one of the promising practical solutions to BB84 QKD with coherent light pulses. In the real world, however, statistical fluctuations with the finite code length cannot be negligible, and the securities of theoretical and experimental researches of the decoy method state QKD so far are based on the asymptotic GLLP's formula which guarantees only that the limit of eavesdropper's information becomes zero as the code length approaches infinity. In this paper, we propose a substantially improved decoy state QKD in the framework of the finite code length and derive the upper bound of eavesdropper's information in the finite code length decoy state QKD with arbitrary number of decoy states of different intensities incorporating the finite statistics. We also show the performance of our decoy QKD and optimal values of parameters by numerical simulation.
The lists of bits processed in quantum key distribution are necessarily of finite length. The need for finite-key unconditional security bounds was recognized long ago, but the theoretical tools have become available only very recently. We provide finite-key unconditional security bounds for two practical implementations of the Bennett–Brassard 1984 coding: prepare-and-measure implementations without decoy states and entanglement-based implementations. A finite-key bound for prepare-and-measure implementations with decoy states is also derived under a simplified treatment of the statistical fluctuations. The presentation is tailored to allow direct application of the bounds in experiments. Finally, the bounds are also evaluated on a priori reasonable expected values of the observed parameters.