Worst case to average case reductions for polynomials

Tali Kaufman, Shachar Lovett

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

38 Scopus citations

Abstract

A degree-d polynomial p inn variables over a field double struk F sign is equidistributed if it takes on each of its |double struk F sign| values close to equally often, and biased otherwise. We say that p has low rank if it can be expressed as a function of a small number of lower degree polynomials. Green and Tao [GT07] have shown that over large fields (i.e when d < |double struk F sign|) a biased polynomial must have low rank. They have also conjectured that bias implies low rank over general fields, but their proof technique fails to show that. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be approximated by a few polynomials of bounded degree (i.e. a polynomial with non negligible correlation with a function of few bounded degree polynomials), can be computed by a few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators. Our work provides another evidence to the structure vs. randomness dichotomy.

Original languageEnglish
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages166-175
Number of pages10
DOIs
StatePublished - 2008
Externally publishedYes
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: 25 Oct 200828 Oct 2008

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Country/TerritoryUnited States
CityPhiladelphia, PA
Period25/10/0828/10/08

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