DROPS

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.69

Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors

Authors: Kuan Cheng and Ruiyang Wu

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract

We study randomness extractors in AC⁰ and NC¹. For the AC⁰ setting, we give a logspace-uniform construction such that for every k ≥ n/poly log n, ε ≥ 2^{-poly log n}, it can extract from an arbitrary (n, k) source, with a small constant fraction entropy loss, and the seed length is O(log n/(ε)). The seed length and output length are optimal up to constant factors matching the parameters of the best polynomial time construction such as [Guruswami et al., 2009]. The range of k and ε almost meets the lower bound in [Goldreich et al., 2015] and [Cheng and Li, 2018]. We also generalize the main lower bound of [Goldreich et al., 2015] for extractors in AC⁰, showing that when k < n/poly log n, even strong dispersers do not exist in non-uniform AC⁰. For the NC¹ setting, we also give a logspace-uniform extractor construction with seed length O(log n/(ε)) and a small constant fraction entropy loss in the output. It works for every k ≥ O(log² n), ε ≥ 2^{-O(√k)}. Our main techniques include a new error reduction process and a new output stretch process, based on low-depth circuit implementations for mergers, condensers, and somewhere extractors.

Cite as

Kuan Cheng and Ruiyang Wu. Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.69,
  author =	{Cheng, Kuan and Wu, Ruiyang},
  title =	{{Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{69:1--69:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.69},
  URN =		{urn:nbn:de:0030-drops-210623},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.69},
  annote =	{Keywords: randomness extractor, uniform AC⁰, error reduction, uniform NC¹, disperser}
}

Search Results

Documents authored by Wu, Ruiyang

Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors

Abstract

Cite as

Thanks for your feedback!

Could not send message