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(log2 N) and extracts N γ/3 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 Ωγ(N1/3) random bits. (As in Zimand's construction, the seed length is O(log2 N) 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(log2 N) and output length Ωγ(N 1/3). 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 Ω(N1/5) output length. Zimand describes a construction with O(logN) seed length and O(2√log N) output length. © 2009 Springer.
CITATION STYLE
De, A., & Trevisan, L. (2009). Extractors using hardness amplification. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5687 LNCS, pp. 462–475). https://doi.org/10.1007/978-3-642-03685-9_35
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