Quantum-Proof Extractors: Optimal up to Constant Factors.

We give the first construction of a family of quantum-proof extractors that has optimal seedlength dependence O(log(n/ǫ)) on the input length n and error ǫ. Our extractors support anymin-entropy k = Ω(log n + log1+α(1/ǫ)) and extract m = (1 − α)k bits that are ǫ-close to uniform,for any desired constant α u003e 0. Previous constructions had a quadratically worse seed length orwere 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 automaticallyquantum-proof, with comparable parameters. The existence of such a reduction forextractors 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 considerhigh entropy sources. We construct quantum-proof extractors with the desired parametersfor such sources by extending a classical approach to extractor construction, based on the use ofblock-sources and sampling, to the quantum setting.Our extractors can be used to obtain improved protocols for device-independent randomnessexpansion and for privacy amplification.