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 + log1+α
(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
