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Physical Review A

Finite-size key in the Bennett 1992 quantum-key-distribution protocol for Rényi entropies

Mhlambululi Mafu, Kevin Garapo, Francesco Petruccione

Published Dec 4, 2013• 15 citations
Publisher

Abstract

A realistic quantum-key-distribution protocol necessarily runs with finite resources. Usually, security proofs for existing quantum key distribution are asymptotic in the sense that certain parameters are exceedingly large compared to practical realistic values. In this paper, we derive bounds on the secret key rates for the Bennett 1992 protocol, which includes a preprocessing step. The derivation for a finite-size key is expressed as an optimization problem by using results from the uncertainty relations and the smooth R\'enyi entropies.

DOI
10.1103/physreva.88.062306

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A realistic quantum-key-distribution protocol necessarily runs with finite resources. Usually, security proofs for existing quantum key distribution are asymptotic in the sense that certain parameters are exceedingly large compared to practical realistic values. In this paper, we derive bounds on the secret key rates for the Bennett 1992 protocol, which includes a preprocessing step. The derivation for a finite-size key is expressed as an optimization problem by using results from the uncertainty relations and the smooth R'enyi entropies.
Finite-size general security proofs for quantum key distribution based on Rényi entropies have recently been introduced. These approaches are more flexible and provide tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. However, deploying them requires minimizing the conditional Rényi entropy, a difficult optimization problem that has hitherto been tackled using ad-hoc techniques based on the Frank-Wolfe algorithm, which are unstable and can only handle particular cases. In this work, we introduce a method based on non-symmetric conic optimization for solving this problem. Our technique is fast, reliable, and completely general. We illustrate its performance on several protocols, whose results represent an improvement over the state of the art.
Quantum key distribution requires tight and reliable bounds on the secret key rate to ensure robust security. This is particularly so for the regime of finite block sizes, where the optimization of generalized Rényi entropic quantities is known to provide tighter bounds on the key rate. However, such an optimization is often non-trivial, and the non-monotonicity of the key rate in terms of the Rényi parameter demands additional optimization to determine the optimal Rényi parameter as a function of block sizes. In this work, we present a tight analytical bound on the Rényi entropy in terms of the Rényi divergence and derive the analytical gradient of the Rényi divergence. This enables us to generalize existing state-of-the-art numerical frameworks for the optimization of the key rate. With this generalized framework, we show improvements in regimes of high loss and low block sizes, which are particularly relevant for long-distance satellite-based protocols.
Quantum key distribution (QKD) is a form of secure communication which transmits secret keys between two trusted parties. Evaluation of the achievable secret key rates are essential to ensure robust security of any QKD protocols. As such, the optimization of the secret key rate to obtain tight and reliable bounds are required especially for instances when the number of transmitted signals are finite. In this work, we make use of the generalized Rényi entropic quantities to obtain a tighter bound on the key rate compared to previous works which uses the von Neumann entropy. However, the optimization of Rényi entropic quantities is often non-trivial. Hence, we derive a generalized bound on the Rényi entropy in terms of the Rényi divergence which allows us to apply the Frank-Wolfe minimization algorithm for the optimization of secret key rates, thus generalizing existing numerical frameworks. With this generalized framework, we show improvements in regimes where finite signals experience high loss, thus ensuring relevance for real-world scenarios, particularly for satellite-based QKD implementations.
We compare three proof techniques for composable finite-size security of quantum key distribution under collective attacks, with emphasis on how the resulting secret-key rates behave at practically relevant block lengths. As a benchmark, we consider the BB84 protocol and evaluate finite-size key-rate estimates obtained from entropic uncertainty relations (EUR), from the asymptotic equipartition property (AEP), and from a direct finite-block analysis based on the conditional min-entropy, which we refer to as the finite-size min-entropy (FME) approach. For BB84 we show that the EUR-based bound provides the most favorable performance across the considered parameter range, while the AEP bound is asymptotically tight but can become overly pessimistic at moderate and small block sizes, where it may fail to certify a positive key. The FME approach remains effective in this small-block regime, yielding nonzero rates in situations where the AEP estimate vanishes, although it is not asymptotically optimal for BB84. These results motivate the use of FME-type analyses for continuous-variable protocols in settings where tight EUR-based bounds are unavailable, notably for coherent-state schemes where current finite-size analyses typically rely on AEP-style corrections.
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.