License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ICALP.2022.10
URN: urn:nbn:de:0030-drops-163519
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Alrabiah, Omar ; Chattopadhyay, Eshan ; Goodman, Jesse ; Li, Xin ; Ribeiro, João

Low-Degree Polynomials Extract From Local Sources

LIPIcs-ICALP-2022-10.pdf (0.9 MB)


We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, this model was introduced by De and Watson (TOCT 2012) and Viola (SICOMP 2014), and is closely related to sources generated by AC⁰ circuits and bounded-width branching programs. In particular, extractors for local sources also work for sources generated by these classical computational models.
Despite being introduced a decade ago, little progress has been made on improving the entropy requirement for extracting from local sources. The current best explicit extractors require entropy n^{1/2}, and follow via a reduction to affine extractors. To start, we prove a barrier showing that one cannot hope to improve this entropy requirement via a black-box reduction of this form. In particular, new techniques are needed.
In our main result, we seek to answer whether low-degree polynomials (over 𝔽₂) hold potential for breaking this barrier. We answer this question in the positive, and fully characterize the power of low-degree polynomials as extractors for local sources. More precisely, we show that a random degree r polynomial is a low-error extractor for n-bit local sources with min-entropy Ω(r(nlog n)^{1/r}), and we show that this is tight.
Our result leverages several new ingredients, which may be of independent interest. Our existential result relies on a new reduction from local sources to a more structured family, known as local non-oblivious bit-fixing sources. To show its tightness, we prove a "local version" of a structural result by Cohen and Tal (RANDOM 2015), which relies on a new "low-weight" Chevalley-Warning theorem.

BibTeX - Entry

  author =	{Alrabiah, Omar and Chattopadhyay, Eshan and Goodman, Jesse and Li, Xin and Ribeiro, Jo\~{a}o},
  title =	{{Low-Degree Polynomials Extract From Local Sources}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-163519},
  doi =		{10.4230/LIPIcs.ICALP.2022.10},
  annote =	{Keywords: Randomness extractors, local sources, samplable sources, AC⁰ circuits, branching programs, low-degree polynomials, Chevalley-Warning}

Keywords: Randomness extractors, local sources, samplable sources, AC⁰ circuits, branching programs, low-degree polynomials, Chevalley-Warning
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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