Better short-seed extractors against quantum knowledge

We construct a strong extractor against quantum storage that works for every min-entropy $k$, has logarithmic seed length, and outputs $\Omega(k)$ bits, provided that the quantum adversary has at most $\beta k$ qubits of memory, for any $\beta < \half$. Previous constructions required poly-logarithmic seed length to output such a fraction of the entropy and, in addition, required super-logarithmic seed length for small values of $k$. 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 extractor against quantum knowledge, in the high guessing entropy regime. Specifically, we construct an extractor that uses a logarithmic seed length and extracts $\Omega(n)$ bits from any source over $\B^n$, provided that the guessing entropy of the source conditioned on the quantum adversary's state is at least $(1-\beta) n$, for any $\beta < \half$. Previous constructions required poly-logarithmic seed length to output $\Omega(n)$ bits from such sources.