Quantum-Proof Extractors: Optimal up to Constant Factors

We give the first construction of a family of quantum-proof extractors that has optimal seed length dependence O(log(n/ǫ)) on the input length n and error ǫ. Our extractors support any min-entropy k = Ω(log n+ log(1/ǫ)) and extract m = (1− α)k bits that are ǫ-close to uniform, for any desired constant α > 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.