Simple extractors via constructions of cryptographic pseudo-random generators

Trevisan has shown that constructions of pseudo-random generators from hard functions (the Nisan-Wigderson approach) also produce extractors. We show that constructions of pseudo-random generators from one-way permutations (the Blum-Micali-Yao approach) can be used for building extractors as well. Using this new technique we build extractors that do not use designs and polynomial-based error-correcting codes and that are very simple and efficient. For example, one extractor produces each output bit separately in $O(\log^2 n)$ time. These extractors work for weak sources with min entropy $\lambda n$, for arbitrary constant $\lambda > 0$, have seed length $O(\log^2 n)$, and their output length is $\approx n^{\lambda/3}$.Comment: 21 pages, an extended abstract will appear in Proc. ICALP 2005; small corrections, some comments and references adde

Randomness Extraction in AC0 and with Small Locality

Randomness extractors, which extract high quality (almost-uniform) random bits from biased random sources, are important objects both in theory and in practice. While there have been significant progress in obtaining near optimal constructions of randomness extractors in various settings, the computational complexity of randomness extractors is still much less studied. In particular, it is not clear whether randomness extractors with good parameters can be computed in several interesting complexity classes that are much weaker than P. In this paper we study randomness extractors in the following two models of computation: (1) constant-depth circuits (AC0), and (2) the local computation model. Previous work in these models, such as [Vio05a], [GVW15] and [BG13], only achieve constructions with weak parameters. In this work we give explicit constructions of randomness extractors with much better parameters. As an application, we use our AC0 extractors to study pseudorandom generators in AC0, and show that we can construct both cryptographic pseudorandom generators (under reasonable computational assumptions) and unconditional pseudorandom generators for space bounded computation with very good parameters. Our constructions combine several previous techniques in randomness extractors, as well as introduce new techniques to reduce or preserve the complexity of extractors, which may be of independent interest. These include (1) a general way to reduce the error of strong seeded extractors while preserving the AC0 property and small locality, and (2) a seeded randomness condenser with small locality.Comment: 62 page

Trevisan's extractor in the presence of quantum side information

Randomness extraction involves the processing of purely classical information and is therefore usually studied in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications, where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information. We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g, extract all the conditional (smooth) min-entropy of the source using a seed of length poly-logarithmic in the input, or only require the seed to be weakly random.Comment: 20+10 pages; v2: extract more min-entropy, use weakly random seed; v3: extended introduction, matches published version with sections somewhat reordere

Short seed extractors against quantum storage

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