Extractors Using Hardness Amplification

• Authors: Anindya De;Luca Trevisan
• Venue: APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
• Year: 2009

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 ***N , his extractor has seed O (log2 N ) and extracts N *** /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.