  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.
