Quantum-Proof Extractors: Optimal up to Constant Factors

TL;DR

This paper presents the first construction of quantum-proof extractors with optimal seed length dependence, supporting sources with high min-entropy, and extends classical methods to the quantum setting, improving cryptographic protocols.

Contribution

It introduces a generic reduction showing classical condensers are quantum-proof, enabling the construction of optimal quantum-proof extractors for high entropy sources.

Findings

Supports any min-entropy $k= ext{Omega}( ext{log}n + ext{log}^{1+eta}(1/ ext{epsilon}))$

Achieves seed length $O( ext{log}(n/ ext{epsilon}))$

Enhances protocols for randomness expansion and privacy amplification

Abstract

We give the first construction of a family of quantum-proof extractors that has optimal seed length dependence $O(\log(n/\varepsilon))$ on the input length $n$ and error $\varepsilon$. Our extractors support any min-entropy $k=\Omega(\log{n} + \log^{1+\alpha}(1/\varepsilon))$ and extract $m=(1-\alpha)k$ bits that are $\varepsilon$-close to uniform, for any desired constant $\alpha > 0$. Previous constructions had a quadratically worse seed length or were restricted to very large input min-entropy or very few output bits. Our result is based on a generic reduction showing that any strong classical condenser is automatically quantum-proof, with comparable parameters. The existence of such a reduction for extractors is a long-standing open question, here we give an affirmative answer for condensers. Once this reduction is established, to obtain our quantum-proof extractors one only needs to consider high entropy sources. We construct quantum-proof extractors with the desired parameters for such sources by extending a classical approach to extractor construction, based on the use of block-sources and sampling, to the quantum setting. Our extractors can be used to obtain improved protocols for device-independent randomness expansion and for privacy amplification.