Constructions of low-degree and error-correcting ε-biased generators

Amir Shpilka*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In this work we give two new constructions of ε-biased generators. Our first construction answers an open question of Dodis and Smith [DS05], and our second construction significantly extends a result of Mossel et al. [MST03]. In particular we obtain the following results: 1. We construct a family of asymptotically good binary codes such that the codes in our family are also ε-biased sets for an exponentially small e. Our encoding and decoding algorithms run in polynomial time in the block length of the code. This answers an open question of Dodis and Smith [DS05]. 2. For every k = o(log n) we construct a degree k ε-biased generator G: {0, 1}m → {0, 1}n (namely, every output bit of the generator is a degree k polynomial in the input bits). For k constant we get that n = Ω(m/log(1/ε))k, which is nearly optimal. Our result also separates degree k generators from generators in NCk0, showing that the stretch of the former can be much larger than the stretch of the latter. The problem of constructing degree k generators was introduced by Mossel et al. [MST03] who gave a construction only for the case of degree 2 generators.

Original languageEnglish
Title of host publicationProceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006
Pages33-45
Number of pages13
DOIs
StatePublished - 2006
Externally publishedYes
Event21st Annual IEEE Conference on Computational Complexity, CCC 2006 - Prague, Czech Republic
Duration: 16 Jul 200620 Jul 2006

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
Volume2006
ISSN (Print)1093-0159

Conference

Conference21st Annual IEEE Conference on Computational Complexity, CCC 2006
Country/TerritoryCzech Republic
CityPrague
Period16/07/0620/07/06

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