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Better short-seed quantum-proof extractors

We construct a strong extractor against quantum storage that works for every min-entropy $k$, has logarithmic seed length, and outputs $Ω(k)$ bits, provided that the quantum adversary has at most $βk$ qubits of memory, for any $β< \half$. The construction works by first condensing the source (with minimal entropy-loss) and then applying an extractor that works well against quantum adversaries when the source is close to uniform. We also obtain an improved construction of a strong quantum-proof extractor in the high min-entropy regime. Specifically, we construct an extractor that uses a logarithmic seed length and extracts $Ω(n)$ bits from any source over $\B^n$, provided that the min-entropy of the source conditioned on the quantum adversary's state is at least $(1-β) n$, for any $β< \half$.