# Finite-key analysis for quantum key distribution with weak coherent pulses based on Bernoulli sampling

## TL;DR

This paper introduces a Bernoulli sampling-based parameter estimation method for finite-key quantum key distribution, improving key rates for BB84 with weak coherent pulses and confirming the finite-key advantage of the DQPS protocol.

## Contribution

It proposes a concise Bernoulli sampling method for finite-key analysis, reducing parameter estimation complexity and enhancing key rates in quantum key distribution protocols.

## Key findings

- Bernoulli sampling improves parameter estimation efficiency.
- The method increases the key rate for BB84 with weak coherent pulses.
- Finite-key security of the DQPS protocol is established, confirming its advantage.

## Abstract

An essential step in quantum key distribution is the estimation of parameters related to the leaked amount of information, which is usually done by sampling of the communication data. When the data size is finite, the final key rate depends on how the estimation process handles statistical fluctuations. Many of the present security analyses are based on the method with simple random sampling, where hypergeometric distribution or its known bounds are used for the estimation. Here we propose a concise method based on Bernoulli sampling, which is related to binomial distribution. Our method is suitable for the BB84 protocol with weak coherent pulses, reducing the number of estimated parameters to achieve a higher key generation rate compared to the method with simple random sampling. We also applied the method to prove the security of the differential-quadrature-phase-shift (DQPS) protocol in the finite-key regime. The result indicates that, the advantage of the DQPS protocol over the phase-encoding BB84 protocol in terms of the key rate, which was previously confirmed in the asymptotic regime, persists in the finite-key regime.