Document Open Access Logo

Extractors for Polynomial Sources over 𝔽₂

Authors Eshan Chattopadhyay , Jesse Goodman , Mohit Gurumukhani



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2024.28.pdf
  • Filesize: 0.87 MB
  • 24 pages

Document Identifiers

Author Details

Eshan Chattopadhyay
  • Cornell University, Ithaca, NY, USA
Jesse Goodman
  • Cornell University, Ithaca, NY, USA
Mohit Gurumukhani
  • Cornell University, Ithaca, NY, USA

Acknowledgements

We want to thank Michael Jaber for helpful discussions.

Cite AsGet BibTex

Eshan Chattopadhyay, Jesse Goodman, and Mohit Gurumukhani. Extractors for Polynomial Sources over 𝔽₂. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.28

Abstract

We explicitly construct the first nontrivial extractors for degree d ≥ 2 polynomial sources over 𝔽₂. Our extractor requires min-entropy k ≥ n - (√{log n})/((log log n / d)^{d/2}). Previously, no constructions were known, even for min-entropy k ≥ n-1. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy k can be generated by O(k) uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below k ≥ n-o(n). In more detail, we show that sumset extractors cannot even disperse from degree 2 polynomial sources with min-entropy k ≥ n-O(n/log log n). In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • Extractors
  • low-degree polynomials
  • varieties
  • sumset extractors

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Omar Alrabiah, Eshan Chattopadhyay, Jesse Goodman, Xin Li, and João Ribeiro. Low-degree polynomials extract from local sources. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 10:1-10:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ICALP.2022.10.
  2. Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, and Avi Wigderson. Simulating independence: New constructions of condensers, ramsey graphs, dispersers, and extractors. J. ACM, 57(4):20:1-20:52, 2010. URL: https://doi.org/10.1145/1734213.1734214.
  3. Eli Ben-Sasson and Ariel Gabizon. Extractors for polynomial sources over fields of constant order and small characteristic. Theory Comput., 9:665-683, 2013. URL: https://doi.org/10.4086/toc.2013.v009a021.
  4. Eli Ben-Sasson and Swastik Kopparty. Affine dispersers from subspace polynomials. SIAM J. Comput., 41(4):880-914, 2012. URL: https://doi.org/10.1137/110826254.
  5. Jean Bourgain. On the construction of affine extractors. GAFA Geometric And Functional Analysis, 17(1):33-57, 2007. Google Scholar
  6. Jean Bourgain, Zeev Dvir, and Ethan Leeman. Affine extractors over large fields with exponential error. Comput. Complex., 25(4):921-931, 2016. URL: https://doi.org/10.1007/s00037-015-0108-5.
  7. Eshan Chattopadhyay and Jesse Goodman. Improved extractors for small-space sources. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 610-621. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00066.
  8. Eshan Chattopadhyay, Jesse Goodman, and Jyun-Jie Liao. Affine extractors for almost logarithmic entropy. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 622-633. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00067.
  9. Eshan Chattopadhyay and Jyun-Jie Liao. Extractors for sum of two sources. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1584-1597. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519963.
  10. Eshan Chattopadhyay and Avishay Tal. Personal communication to li and zuckerman. Google Scholar
  11. Gil Cohen and Avishay Tal. Two structural results for low degree polynomials and applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24-26, 2015, Princeton, NJ, USA, volume 40 of LIPIcs, pages 680-709. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.680.
  12. Anindya De and Thomas Watson. Extractors and lower bounds for locally samplable sources. ACM Trans. Comput. Theory, 4(1):3:1-3:21, 2012. URL: https://doi.org/10.1145/2141938.2141941.
  13. Matt DeVos and Ariel Gabizon. Simple affine extractors using dimension expansion. In Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC 2010, Cambridge, Massachusetts, USA, June 9-12, 2010, pages 50-57. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/CCC.2010.14.
  14. Zeev Dvir. Extractors for varieties. Comput. Complex., 21(4):515-572, 2012. URL: https://doi.org/10.1007/s00037-011-0023-3.
  15. Zeev Dvir, Ariel Gabizon, and Avi Wigderson. Extractors and rank extractors for polynomial sources. Comput. Complex., 18(1):1-58, 2009. URL: https://doi.org/10.1007/s00037-009-0258-4.
  16. Ariel Gabizon and Ran Raz. Deterministic extractors for affine sources over large fields. Comb., 28(4):415-440, 2008. URL: https://doi.org/10.1007/s00493-008-2259-3.
  17. Alexander Golovnev and Alexander S. Kulikov. Weighted gate elimination: Boolean dispersers for quadratic varieties imply improved circuit lower bounds. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 405-411. ACM, 2016. URL: https://doi.org/10.1145/2840728.2840755.
  18. Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit depth reductions. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.24.
  19. Zeyu Guo, Ben Lee Volk, Akhil Jalan, and David Zuckerman. Extractors for images of varieties. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 46-59. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585109.
  20. Johan Håstad, Russell Impagliazzo, Leonid A Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM Journal on Computing, 28(4):1364-1396, 1999. Preliminary version in STOC 1989. Google Scholar
  21. Miguel Herrero-Collantes and Juan Carlos Garcia-Escartin. Quantum random number generators. Reviews of Modern Physics, 89(1):015004, 2017. Google Scholar
  22. Jesse Kamp, Anup Rao, Salil Vadhan, and David Zuckerman. Deterministic extractors for small-space sources. Journal of Computer and System Sciences, 77(1):191-220, 2011. Preliminary version in STOC 2006. Google Scholar
  23. Swastik Kopparty. Algebraic methods in randomness and pseudorandomness. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2010. URL: https://hdl.handle.net/1721.1/62425.
  24. Fu Li and David Zuckerman. Improved extractors for recognizable and algebraic sources. In 23rd International Conference on Randomization and Computation (RANDOM), 2019. Google Scholar
  25. Xin Li. A new approach to affine extractors and dispersers. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, USA, June 8-10, 2011, pages 137-147. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/CCC.2011.27.
  26. Xin Li. Improved two-source extractors, and affine extractors for polylogarithmic entropy. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 168-177. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.26.
  27. Xin Li. Two source extractors for asymptotically optimal entropy, and (many) more. In 64th Annual Symposium on Foundations of Computer Science (FOCS 2023). IEEE Computer Society, 2023. Google Scholar
  28. Rajeev Motwani and Prabhakar Raghavan. Randomized algorithms. Cambridge University Press, 1995. Google Scholar
  29. Anup Rao. Randomness extractors for independent sources and applications. PhD thesis, The University of Texas at Austin, 2007. Google Scholar
  30. Anup Rao. Extractors for low-weight affine sources. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15-18 July 2009, pages 95-101. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/CCC.2009.36.
  31. Razbarov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987. Google Scholar
  32. Maximus Redman, Lauren Rose, and Raphael Walker. A small maximal sidon set in ℤ₂ⁿ. SIAM J. Discret. Math., 36(3):1861-1867, 2022. URL: https://doi.org/10.1137/21m1454663.
  33. Zachary Remscrim. The hilbert function, algebraic extractors, and recursive fourier sampling. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 197-208. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.29.
  34. Ronen Shaltiel. Dispersers for affine sources with sub-polynomial entropy. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 247-256. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.37.
  35. R. Smolensky. On representations by low-degree polynomials. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 130-138, 1993. URL: https://doi.org/10.1109/SFCS.1993.366874.
  36. Luca Trevisan and Salil P. Vadhan. Extracting randomness from samplable distributions. In 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, 12-14 November 2000, Redondo Beach, California, USA, pages 32-42. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/SFCS.2000.892063.
  37. Salil P. Vadhan. Pseudorandomness. Found. Trends Theor. Comput. Sci., 7(1-3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  38. Emanuele Viola. Extractors for circuit sources. SIAM J. Comput., 43(2):655-672, 2014. URL: https://doi.org/10.1137/11085983X.
  39. Amir Yehudayoff. Affine extractors over prime fields. Comb., 31(2):245-256, 2011. URL: https://doi.org/10.1007/s00493-011-2604-9.
  40. S Znám. On a combinatorical problem of k. zarankiewicz. In Colloquium Mathematicum, volume 11, pages 81-84. Instytut Matematyczny Polskiej Akademii Nauk, 1963. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail