Abstract
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 \(\tilde \Omega_\gamma (N^{1/3}) \) 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 \(\tilde \Omega_\gamma (N^{1/3})\).
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 \(\tilde \Omega (N^{1/5})\) output length. Zimand describes a construction with O(logN) seed length and \(O(2^{\sqrt{\log N}})\) output length.
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De, A., Trevisan, L. (2009). Extractors Using Hardness Amplification. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_35
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