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Low-Degree Polynomials Are Good Extractors

Authors Omar Alrabiah , Jesse Goodman , Jonathan Mosheiff , João Ribeiro

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LIPIcs.APPROX-RANDOM.2025.38.pdf
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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • randomness extractors
  • low-degree polynomials
  • local sources
  • polynomial sources
  • variety sources
  • sumset sources
  • sumset extractors
  • Reed-Muller codes
  • lower bounds

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Abstract

We prove that random low-degree polynomials (over 𝔽₂) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, (2) affine sources, and (3) local sources. We significantly generalize these results, and prove the following.  
1) Low-degree polynomials extract from small families. We show that a random low-degree polynomial is a good low-error extractor for any small family of sources. In particular, we improve the positive result of Alrabiah, Chattopadhyay, Goodman, Li, and Ribeiro (ICALP 2022) for local sources, and give new results for polynomial and variety sources via a single unified approach.
2) Low-degree polynomials extract from sumset sources. We show that a random low-degree polynomial is a good extractor for sumset sources, which are the most general large family of sources (capturing independent sources, interleaved sources, small-space sources, and more). Formally, for any even d, we show that a random degree d polynomial is an ε-error extractor for n-bit sumset sources with min-entropy k = O(d(n/ε²)^{2/d}). This is nearly tight in the polynomial error regime. 
Our results on sumset extractors imply new complexity separations for linear ROBPs, and the tools that go into its proof may be of independent interest. The two main tools we use are a new structural result on sumset-punctured Reed-Muller codes, paired with a novel type of reduction between extractors. Using the new structural result, we obtain new limits on the power of sumset extractors, strengthening and generalizing the impossibility results of Chattopadhyay, Goodman, and Gurumukhani (ITCS 2024).

Cite As Get BibTex

Omar Alrabiah, Jesse Goodman, Jonathan Mosheiff, and João Ribeiro. Low-Degree Polynomials Are Good Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 38:1-38:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.38

Author Details

Omar Alrabiah
  • University of California, Berkeley, CA, USA
Jesse Goodman
  • The University of Texas at Austin, TX, USA
Jonathan Mosheiff
  • Ben-Gurion University of the Negev, Be'er-Sheva, Israel
João Ribeiro
  • Instituto de Telecomunicações and Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal

Funding

  • Alrabiah, Omar: Supported by a Saudi Arabian Cultural Mission (SACM) Scholarship, NSF Award CCF-2210823, and a Simons Investigator Award (Venkatesan Guruswami).
  • Goodman, Jesse: Supported by a Simons Investigator Award (#409864, David Zuckerman).
  • Mosheiff, Jonathan: Supported by Israel Science Foundation grant 3450/24, an Alon Fellowship, and DOE grant #DE-SC0024124.
  • Ribeiro, João: Supported by NOVA LINCS (ref. UIDB/04516/2020), FCT/MECI through national funds and when applicable co-funded EU funds under UID/50008: Instituto de Telecomunicações, and DOE grant #DE-SC0024124.

Acknowledgements

We thank Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, and Chao Yan for sharing with us an early draft of their work. We also thank Dean Doron for insightful discussions and feedback, and Mohit Gurumukhani for helpful pointers to facts about quadratic forms. Part of this work was carried out while the authors were visiting the Simons Institute for the Theory of Computing at UC Berkeley, and while the second and fourth authors were visiting NTT Research in Sunnyvale, CA. The fourth author was based at NOVA LINCS and NOVA School of Science and Technology for most of the duration of this work.

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