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 $\mathrm{\Omega }(k)$ bits, provided that the quantum adversary has at most $\beta k$ qubits of memory, for any $\beta <\text{half}$. 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 $\mathrm{\Omega }(n)$ bits from any source over ${\text{B}}^n$, provided that the min-entropy of the source conditioned on the quantum adversary’s state is at least $(1-\beta )n$, for any $\beta <\text{half}$.