We give near-optimal constructions of extractors secure against quantum bounded-storage adversaries. One instantiation gives the first such extractor to achieve an output length Theta(K-b), where K is the source's entropy and b the adversary's storage, depending linearly on the adversary's amount of storage, together with a poly-logarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length Theta((K-b)/K^g) for any g>0. In contrast, the previous best construction [Ta-Shma, STOC'09] could only extract (K/b)^(1/15) bits.

Our construction follows Trevisan's general reconstruction paradigm, and in fact our proof of security shows that essentially all extractors constructed using this paradigm are secure against quantum storage, with optimal parameters. Our argument is based on bounds for a generalization of quantum random access codes, which we call quantum functional access codes. This is crucial as it lets us avoid the local list-decoding algorithm central to the approach in [Ta-Shma, STOC'09] which was the source of the multiplicative overhead.

Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications.