Document Open Access Logo

Extractor Lower Bounds, Revisited

Authors Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro , Noah Stephens-Davidowitz

Thumbnail PDF


  • Filesize: 0.54 MB
  • 20 pages

Document Identifiers

Author Details

Divesh Aggarwal
  • National University of Singapore, Singapore
Siyao Guo
  • New York University Shanghai, China
Maciej Obremski
  • National University of Singapore, Singapore
João Ribeiro
  • Imperial College London, UK
Noah Stephens-Davidowitz
  • Cornell University, Ithaca, NY, USA


The authors wish to thank the anonymous reviewers for several comments that improved the paper.

Cite AsGet BibTex

Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro, and Noah Stephens-Davidowitz. Extractor Lower Bounds, Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 1:1-1:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a "change in quantifiers" over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input sources with sufficient min-entropy, a somewhere extractor only requires that there exists a seed whose output bias is small. More generally, we study what we call probable extractors, which on input a source with sufficient min-entropy guarantee that a large enough fraction of seeds have small enough associated output bias. Such extractors have played a key role in many constructions of pseudorandom objects, though they are often defined implicitly and have not been studied extensively. Prior known techniques fail to yield good seed length lower bounds when applied to the variants above. Our novel approach yields significantly improved lower bounds for somewhere and probable extractors. To complement this, we construct a somewhere extractor that implies our lower bound for such functions is tight in the high min-entropy regime. Surprisingly, this means that a random function is far from an optimal somewhere extractor in this regime. The techniques that we develop also yield an alternative, simpler proof of the celebrated optimal lower bound for strong extractors originally due to Radhakrishnan and Ta-Shma (SIAM J. Discrete Math., 2000).

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
  • randomness extractors
  • lower bounds
  • explicit constructions


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


  1. Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro, and Noah Stephens-Davidowitz. Extractor lower bounds, revisited. Electronic Colloquium on Computational Complexity (ECCC), 26(173), 2019. URL:
  2. Divesh Aggarwal, Maciej Obremski, João Ribeiro, Luisa Siniscalchi, and Ivan Visconti. How to extract useful randomness from unreliable sources. In Anne Canteaut and Yuval Ishai, editors, Advances in Cryptology - EUROCRYPT 2020, pages 343-372, Cham, 2020. Springer International Publishing. Google Scholar
  3. Ziv Bar-Yossef, Ravi Kumar, and D. Sivakumar. Sampling algorithms: Lower bounds and applications. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC ’01, page 266–275, New York, NY, USA, 2001. Association for Computing Machinery. URL:
  4. 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, May 2010. URL:
  5. Boaz Barak, Anup Rao, Ronen Shaltiel, and Avi Wigderson. 2-source dispersers for n^o(1) entropy, and Ramsey graphs beating the Frankl-Wilson construction. Annals of Mathematics, pages 1483-1543, 2012. Google Scholar
  6. Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, and Amnon Ta-Shma. A new approach for constructing low-error, two-source extractors. In Proceedings of the 33rd Computational Complexity Conference, CCC '18, pages 3:1-3:19, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  7. Eshan Chattopadhyay, Vipul Goyal, and Xin Li. Non-malleable extractors and codes, with their many tampered extensions. In Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, STOC '16, pages 285-298, New York, NY, USA, 2016. ACM. URL:
  8. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. Annals of Mathematics, 189(3):653-705, 2019. URL:
  9. Gil Cohen. Local correlation breakers and applications to three-source extractors and mergers. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 845-862, October 2015. URL:
  10. Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, and Madhu Sudan. Extensions to the method of multiplicities, with applications to Kakeya sets and mergers. SIAM Journal on Computing, 42(6):2305-2328, 2013. URL:
  11. Zeev Dvir and Ran Raz. Analyzing linear mergers. Random Structures & Algorithms, 32(3):334-345, 2008. URL:
  12. Zeev Dvir and Amir Shpilka. An improved analysis of linear mergers. computational complexity, 16(1):34-59, May 2007. URL:
  13. Zeev Dvir and Avi Wigderson. Kakeya sets, new mergers, and old extractors. SIAM Journal on Computing, 40(3):778-792, 2011. URL:
  14. Philip Klein and Neal E. Young. On the number of iterations for Dantzig-Wolfe optimization and packing-covering approximation algorithms. SIAM Journal on Computing, 44(4):1154-1172, 2015. URL:
  15. Xin Li. Improved constructions of three source extractors. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 126-136, June 2011. URL:
  16. Xin Li. Extractors for a constant number of independent sources with polylogarithmic min-entropy. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 100-109, October 2013. URL:
  17. Xin Li. New independent source extractors with exponential improvement. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 783-792, New York, NY, USA, June 2013. ACM. URL:
  18. Xin Li. Three-source extractors for polylogarithmic min-entropy. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 863-882, October 2015. URL:
  19. Xin Li. Improved two-source extractors, and affine extractors for polylogarithmic entropy. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 168-177, October 2016. URL:
  20. Chi-Jen Lu, Omer Reingold, Salil Vadhan, and Avi Wigderson. Extractors: Optimal up to constant factors. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 602-611, New York, NY, USA, 2003. ACM. URL:
  21. Noam Nisan and David Zuckerman. Randomness is linear in space. Journal of Computer and System Sciences, 52(1):43-52, 1996. URL:
  22. Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1):2-24, 2000. URL:
  23. Anup Rao. An exposition of Bourgain’s 2-source extractor. Electronic Colloquium on Computational Complexity (ECCC), 14(034), 2007. URL:
  24. Anup Rao. Extractors for a constant number of polynomially small min-entropy independent sources. SIAM Journal on Computing, 39(1):168-194, 2009. URL:
  25. Ran Raz. Extractors with weak random seeds. In Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC '05, pages 11-20, New York, NY, USA, 2005. ACM. URL:
  26. Amnon Ta-Shma. On extracting randomness from weak random sources (extended abstract). In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 276-285, New York, NY, USA, 1996. ACM. URL:
  27. David Zuckerman. Randomness-optimal oblivious sampling. Random Structures & Algorithms, 11(4):345-367, 1997. URL:<345::AID-RSA4>3.0.CO;2-Z.
  28. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 681-690, New York, NY, USA, 2006. ACM. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail