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2008
Springer

Hardness Amplification via Space-Efficient Direct Products

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Hardness Amplification via Space-Efficient Direct Products
We prove a version of the derandomized Direct Product lemma for deterministic space-bounded algorithms. Suppose a Boolean function g : {0, 1}n {0, 1} cannot be computed on more than a fraction 1 - of inputs by any deterministic time T and space S algorithm, where 1/t for some t. Then for t-step walks w = (v1, . . . , vt) in some explicit d-regular expander graph on 2n vertices, the function g (w) def = (g(v1), . . . , g(vt)) cannot be computed on more than a fraction 1 - (t) of inputs by any deterministic time T/dt - poly(n) and space S - O(t) algorithm. As an application, by iterating this construction, we get a deterministic linear-space "worst-case to constant average-case" hardness amplification reduction, as well as a family of logspace encodable/decodable error-correcting codes that can correct up to a constant fraction of errors. Logspace encodable/decodable codes (with linear-time encoding and decoding) were previously constructed by Spielman (1996). Our codes ha...
Venkatesan Guruswami, Valentine Kabanets
Added 09 Dec 2010
Updated 09 Dec 2010
Type Journal
Year 2008
Where CC
Authors Venkatesan Guruswami, Valentine Kabanets
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