P=BPP unless E has sub-exponential circuits: Derandomizing the XOR Lemma (Preliminary Version) (1996)
| Venue: | In Proceedings of the 29th STOC |
| Citations: | 41 - 6 self |
BibTeX
@INPROCEEDINGS{Impagliazzo96p=bppunless,
author = {Russell Impagliazzo and Avi Wigderson},
title = {P=BPP unless E has sub-exponential circuits: Derandomizing the XOR Lemma (Preliminary Version)},
booktitle = {In Proceedings of the 29th STOC},
year = {1996},
pages = {220--229},
publisher = {ACM Press}
}
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Abstract
Yao showed that the XOR of independent random instances of a somewhat hard Boolean function becomes almost completely unpredictable. In this paper we show that, in non-uniform settings, total independence is not necessary for this result to hold. We give a pseudo-random generator which produces n instances of the function for which the analog of the XOR lemma holds. This is the first derandomization of a "direct product" result. Our generator is a combination of two known ones - the random walks on expander graphs of [1, 9, 19] and the nearly disjoint subsets generator of [23]. The quality of the generator is proved via a new proof of the XOR lemma, which might also be useful for other direct product results. Combining our generator with the approach of [25, 6] and the generator of [16] gives substantially improved results for hardness vs. randomness trade-offs. In particular, we show that if any problem in E = DT IME(2 O(n) ) has circuit complexity 2\Omega\Gamma n) , the...
Keyphrases
xor lemma sub-exponential circuit preliminary version direct product result disjoint subset generator hard boolean function pseudo-random generator first derandomization randomness trade-off random walk dt ime expander graph new proof total independence omega gamma non-uniform setting hardness v known one independent random instance
