TY - GEN

T1 - Worst case to average case reductions for polynomials

AU - Kaufman, Tali

AU - Lovett, Shachar

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=57949106800&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.17

DO - 10.1109/FOCS.2008.17

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:57949106800

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 166

EP - 175

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -