Extractors Using Hardness Amplification

Zimand [24] presented simple constructions of locally computable strong extractors whose analysis relies on the direct product theorem for one-way functions and on the Blum-Micali-Yao generator. For N -bit sources of entropy γN , his extractor has seed O(logN) and extracts N random bits. We show that his construction can be analyzed based solely on the direct product theorem for general functions. Using the direct product theorem of Impagliazzo et al. [6], we show that Zimand’s construction can extract Ω̃γ(N ) random bits. (As in Zimand’s construction, the seed length is O(logN) bits.) We also show that a simplified construction can be analyzed based solely on the XOR lemma. Using Levin’s proof of the XOR lemma [8], we provide an alternative simpler construction of a locally computable extractor with seed length O(logN) and output length Ω̃γ(N ). Finally, we show that the derandomized direct product theorem of Impagliazzo and Wigderson [7] can be used to derive a locally computable extractor construction with O(logN) seed length and Ω̃(N) output length. Zimand describes a construction with O(logN) seed length and O(2 √ logN ) output length.