TY - GEN

T1 - Extractors for Images of Varieties

AU - Guo, Zeyu

AU - Volk, Ben Lee

AU - Jalan, Akhil

AU - Zuckerman, David

N1 - Publisher Copyright:
© 2023 ACM.

PY - 2023/6/2

Y1 - 2023/6/2

N2 - We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map f : Fqr → Fqn to an element sampled uniformly at random from a k-dimensional variety V † Fqr. This class of sources generalizes both polynomial sources, studied by Dvir, Gabizon and Wigderson (FOCS 2007, Comput. Complex. 2009), and variety sources, studied by Dvir (CCC 2009, Comput. Complex. 2012). Assuming certain natural non-degeneracy conditions on the map f and the variety V, which in particular ensure that the source has enough min-entropy, we extract almost all the min-entropy of the distribution. Unlike the Dvir-Gabizon-Wigderson and Dvir results, our construction works over large enough finite fields of arbitrary characteristic. One key part of our construction is an improved deterministic rank extractor for varieties. As a by-product, we obtain explicit Noether normalization lemmas for affine varieties and affine algebras. Additionally, we generalize a construction of affine extractors with exponentially small error due to Bourgain, Dvir and Leeman (Comput. Complex. 2016) by extending it to all finite prime fields of quasipolynomial size.

AB - We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map f : Fqr → Fqn to an element sampled uniformly at random from a k-dimensional variety V † Fqr. This class of sources generalizes both polynomial sources, studied by Dvir, Gabizon and Wigderson (FOCS 2007, Comput. Complex. 2009), and variety sources, studied by Dvir (CCC 2009, Comput. Complex. 2012). Assuming certain natural non-degeneracy conditions on the map f and the variety V, which in particular ensure that the source has enough min-entropy, we extract almost all the min-entropy of the distribution. Unlike the Dvir-Gabizon-Wigderson and Dvir results, our construction works over large enough finite fields of arbitrary characteristic. One key part of our construction is an improved deterministic rank extractor for varieties. As a by-product, we obtain explicit Noether normalization lemmas for affine varieties and affine algebras. Additionally, we generalize a construction of affine extractors with exponentially small error due to Bourgain, Dvir and Leeman (Comput. Complex. 2016) by extending it to all finite prime fields of quasipolynomial size.

KW - Extractors

KW - Polynomial Maps

KW - Pseudorandomness

KW - Varieties

UR - http://www.scopus.com/inward/record.url?scp=85163102545&partnerID=8YFLogxK

U2 - 10.1145/3564246.3585109

DO - 10.1145/3564246.3585109

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AN - SCOPUS:85163102545

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 46

EP - 59

BT - STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing

A2 - Saha, Barna

A2 - Servedio, Rocco A.

PB - Association for Computing Machinery

T2 - 55th Annual ACM Symposium on Theory of Computing, STOC 2023

Y2 - 20 June 2023 through 23 June 2023

ER -