AC0中具有小局部性的随机性提取
Randomness Extraction in AC0 and with Small Locality
摘要：
随机抽取器是从有偏随机源中抽取高质量（几乎均匀）随机比特的一种方法，在理论和实践中都有重要意义。虽然在各种设置中获得随机性提取器的接近最优的构造方面已经有了显著的进展，但是对随机性提取器的计算复杂性的研究仍然少得多。特别是，目前还不清楚是否可以计算的随机性提取器与良好的参数在几个有趣的复杂性类，比P弱得多。在本文中，我们研究随机性提取器在以下两个模型的计算：（1）恒定深度电路（AC 0），和（2）本地计算模型。这些模型中以前的工作，如[Vio05a]，[GVW15]和[BG13]，只实现了弱参数的构造。在这项工作中，我们给出了具有更好参数的随机性提取器的显式构造。作为应用，我们使用我们的AC0提取器来研究AC0中的伪随机生成器，并表明我们可以构造密码伪随机生成器（在合理的计算假设下）和具有非常好的参数的空间有界计算的无条件伪随机生成器。我们的构造结合了随机性提取器中的几种先前技术，并引入了新技术来降低或保留提取器的复杂性，这可能是独立的兴趣。这些包括（1）一种通用的方法来减少强种子提取器的错误，同时保持AC 0属性和小局部性，以及（2）具有小局部性的种子随机性冷凝器。
英文摘要：
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.
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