Let C be a class of probability distributions over a finite set Ω. A function D: Ω ↦ → {0, 1} m is a disperser for C with entropy threshold k and error ɛ if for any distribution X in C such that X gives positive probability to at least 2k elements we have that the distribution D(X) gives positive probability to at least (1 − ɛ)2m elements. A long line of research is devoted to giving explicit (that is polynomial time computable) dispersers (and related objects called “extractors”) for various classes of distributions while trying to maximize m as a function of k. In this paper we are interested in explicitly constructing zero-error dispersers (that is dispersers with error ɛ = 0). For several interesting classes of distributions there are explicit constructions in the literature of zero-error dispersers with “small ” output length m and we give improved constructions that achieve “large ” output length, namely m = Ω(k). We achieve this by developing a general technique to improve the output length of zero-error dispersers (namely, to transform a disperser with short output length into one with large output length). This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given in [31] building on earlier work

Some, but not all, extractors resist adversaries with limited quantum storage. In this paper we show that Trevisan’s extractor has this property, thereby showing an extractor against quantum storage with logarithmic seed length. 1

Abstract. In this paper we show that a construction of Trevisan, solving the privacy amplification problem in the classical setting, also solves the problem when the adversary may keep 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.

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 < 1 2. 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 extractsΩ(n) bits from any source over {0,1} n, provided that the min-entropy of the source conditioned on the quantum adversary’s state is at least (1−β)n, for anyβ < 1 2. 1