Document Open Access Logo

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

Authors Kuan Cheng , Ruiyang Wu

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2024.69.pdf
  • Filesize: 0.91 MB
  • 22 pages

Document Identifiers

Author Details

Kuan Cheng
  • CFCS, School of CS, Peking University, China
Ruiyang Wu
  • CFCS, School of CS, Peking University, China

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.69

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • randomness extractor
  • uniform AC⁰
  • error reduction
  • uniform NC¹
  • disperser

Metrics

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

Related Versions

References

  1. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  2. Z. Bar-Yossef, O. Reingold, R. Shaltiel, and L. Trevisan. Streaming computation of combinatorial objects. In Proceedings of the 17th Annual IEEE Conference on Computational Complexity, pages 165-174, 2002. Google Scholar
  3. David A Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in nc. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 1-5, 1986. Google Scholar
  4. Paul Beame, Stephen A. Cook, and H. James Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15(4):994-1003, 1986. URL: https://doi.org/10.1137/0215070.
  5. J. L. Carter and M. N. Wegman. Universal classes of hash functions. Journal of Computer and System Sciences, 18:143-154, 1979. Google Scholar
  6. Lijie Chen and Roei Tell. Hardness vs randomness, revised: Uniform, non-black-box, and instance-wise. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 125-136. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00021.
  7. Kuan Cheng and Xin Li. Randomness extraction in ac0 and with small locality. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM) 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  8. Gil Cohen, Dean Doron, Ori Sberlo, and Amnon Ta-Shma. Approximating iterated multiplication of stochastic matrices in small space. 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 35-45. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585181.
  9. 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, January 2013. URL: https://doi.org/10/f5msx6.
  10. Oded Goldreich, Emanuele Viola, and Avi Wigderson. On randomness extraction in AC0. In Proceedings of the 30th Conference on Computational Complexity, CCC '15, pages 601-668, Dagstuhl, DEU, June 2015. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  11. Oded Goldreich and Avi Wigderson. Tiny families of functions with random properties (preliminary version) a quality-size trade-off for hashing. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 574-584, 1994. Google Scholar
  12. Shafi Goldwasser, Dan Gutfreund, Alexander Healy, Tali Kaufman, and Guy N. Rothblum. Verifying and decoding in constant depth. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 440-449. ACM, 2007. Google Scholar
  13. Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. Journal of the ACM, 56(4):1-34, June 2009. URL: https://doi.org/10/ctbzhm.
  14. Alexander Healy and Emanuele Viola. Constant-Depth circuits for arithmetic in finite fields of characteristic two. In Proceedings of the 23rd Annual Conference on Theoretical Aspects of Computer Science, STACS'06, pages 672-683, Berlin, Heidelberg, February 2006. Springer-Verlag. URL: https://doi.org/10/df5dfs.
  15. Alexander D Healy. Randomness-efficient sampling within nc1. Computational Complexity, 17(1):3-37, 2008. Google Scholar
  16. Russel Impagliazzo, Leonid A. Levin, and Michael Luby. Pseudo-random generation from one-way functions. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 12-24, 1989. Google Scholar
  17. Russell Impagliazzo. Hard-core distributions for somewhat hard problems. In Foundations of Computer Science, 1995. Proceedings., 36th Annual Symposium on, pages 538-545. IEEE, 1995. Google Scholar
  18. Russell Impagliazzo and Avi Wigderson. P=BPP unless E has sub-exponential circuits: Derandomizing the xor lemma. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997. Google Scholar
  19. Itay Kalev and Amnon Ta-Shma. Unbalanced expanders from multiplicity codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM) 2022, volume 245 of LIPIcs, pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2022.12.
  20. Chi-Jen Lu, Omer Reingold, Salil Vadhan, and Avi Wigderson. Extractors: Optimal up to Constant Factors. Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 2003. URL: https://doi.org/10/bw2j9d.
  21. Noam Nisan and Amnon Ta-Shma. Extracting randomness: A survey and new constructions. Journal of Computer and System Sciences, 58:148-173, 1999. Google Scholar
  22. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, October 1994. Google Scholar
  23. Noam Nisan and David Zuckerman. Randomness is linear in space. Journal of Computer and System Sciences, 52(1):43-52, 1996. Google Scholar
  24. Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors and depth-two superconcentrators. Siam Journal on Discrete Mathematics, 13:2-24, 2000. Google Scholar
  25. Ran Raz, Omer Reingold, and Salil P. Vadhan. Error reduction for extractors. In 40th Annual Symposium on Foundations of Computer Science, FOCS '99, pages 191-201. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814591.
  26. Ran Raz, Omer Reingold, and Salil P. Vadhan. Extracting all the randomness and reducing the error in trevisan’s extractors. J. Comput. Syst. Sci., 65(1):97-128, 2002. URL: https://doi.org/10.1006/JCSS.2002.1824.
  27. Omer Reingold, Ronen Shaltiel, and Avi Wigderson. Extracting randomness via repeated condensing. In 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, pages 22-31. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/SFCS.2000.892008.
  28. Benjamin Rossman. An entropy proof of the switching lemma and tight bounds on the decision-tree size of AC0, 2017. URL: https://users.cs.duke.edu/~br148/logsize.pdf.
  29. Ronen Shaltiel. Recent developments in explicit constructions of extractors. Bulletin of the European Association for Theoretical Computer Science, 77:67-95, 2002. Google Scholar
  30. Ronen Shaltiel. An introduction to randomness extractors. In Proceedings of the 38th International Colloquium on Automata, Languages, and Programming, 2011. Google Scholar
  31. A. Srinivasan and D. Zuckerman. Computing with very weak random sources. SIAM Journal on Computing, 28:1433-1459, 1999. Google Scholar
  32. Amnon Ta-Shma. On extracting randomness from weak random sources. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 276-285, 1996. Google Scholar
  33. Amnon Ta-Shma. Almost optimal dispersers. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, 1998, pages 196-202. ACM, 1998. URL: https://doi.org/10.1145/276698.276736.
  34. Amnon Ta-Shma and Christopher Umans. Better condensers and new extractors from parvaresh-vardy codes. In 2012 IEEE 27th Conference on Computational Complexity, pages 309-315. IEEE, 2012. Google Scholar
  35. Roei Tell. Improved bounds for quantified derandomization of constant-depth circuits and polynomials. computational complexity, 28(2):259-343, 2019. Google Scholar
  36. Luca Trevisan. Extractors and pseudorandom generators. Journal of the ACM, 48(4):860-879, 2001. URL: https://doi.org/10.1145/502090.502099.
  37. Salil Vadhan. The unified theory of pseudorandomness. SIGACT News, 38, 2007. Google Scholar
  38. Salil Vadhan. Pseudorandomness. Foundations and Trendsregistered in Theoretical Computer Science, 7(1-3):1-336, 2012. Google Scholar
  39. Emanuele Viola. The complexity of constructing pseudorandom generators from hard functions. computational complexity, 13(3-4):147-188, 2005. Google Scholar
  40. Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125-138, 1999. Google Scholar
  41. David Zuckerman. Randomness-optimal oblivious sampling. Random Structures and Algorithms, 11(4):345-367, December 1997. URL: https://doi.org/10/cr8kht.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact