Short seed extractors against quantum storage

In the classical privacy amplification problem Alice and Bob share information that is only partially secret towards an eavesdropper Charlie. Their goal is to distill this information to a shorter string that is completely secret. The classical privacy amplification problem can be solved almost optimally using extractors. An interesting variant of the problem, where the eavesdropper Charlie is allowed to keep quantum information rather than just classical information, was introduced by Konig, Maurer and Renner. In this setting, the eavesdropper Charlie may entangle himself with the input (without changing it) and the only limitation Charlie has is that it may keep at most b qubits of storage. A natural question is whether there are classical extractors that are good even against quantum storage. Recent work has shown that some classical extractors miserably fail against quantum storage. At the same time, it was shown that some other classical extractors work well even against quantum storage, but all these extractors had a large seed length that was either as large as the extractor output, or as large as the quantum storage available to the eavesdropper. In this paper we show that a modified version of Trevisan's extractor is good even against quantum storage, thereby giving the first such construction with logarithmic seed length. The technique we use is a combination of Trevisan's approach of constructing an extractor from a black-box pseudorandom generator, together with locally list-decodable codes and previous work done on quantum random access codes.