Fractional Pseudorandom Generators from Any Fourier Level

Authors Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, Abhishek Shetty



PDF
Thumbnail PDF

File

LIPIcs.CCC.2021.10.pdf
  • Filesize: 0.8 MB
  • 24 pages

Document Identifiers

Author Details

Eshan Chattopadhyay
  • Department of Computer Science, Cornell University, Ithaca, NY, USA
Jason Gaitonde
  • Department of Computer Science, Cornell University, Ithaca, NY, USA
Chin Ho Lee
  • Department of Computer Science, Columbia University, New York City, NY, USA
Shachar Lovett
  • Department of Computer Science, University of California, San Diego, CA, USA
Abhishek Shetty
  • Department of Computer Science, University of California, Berkeley, CA, USA

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions.

Cite AsGet BibTex

Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty. Fractional Pseudorandom Generators from Any Fourier Level. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CCC.2021.10

Abstract

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2019] that exploit L₁ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [Chattopadhyay et al., 2019], or have polynomial dependence on the error parameter in the seed length [Eshan Chattopadhyay et al., 2019], and thus answers an open question in [Eshan Chattopadhyay et al., 2019]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [Chattopadhyay et al., 2019], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L₁ notion in previous works. By generalizing a connection established in [Chattopadhyay et al., 2020], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for 𝔽₂ polynomials with seed length close to the state-of-the-art construction due to Viola [Emanuele Viola, 2009].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • pseudorandomness
  • pseudorandom generators
  • Fourier analysis

Metrics

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

References

  1. Rohit Agrawal. Coin theorems and the Fourier expansion. Chicago Journal of Theoretical Computer Science, 2020(4), August 2020. Google Scholar
  2. Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf. Improved bounds on Fourier entropy and min-entropy. In Christophe Paul and Markus Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, March 10-13, 2020, Montpellier, France, volume 154 of LIPIcs, pages 45:1-45:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.45.
  3. Nikhil Bansal and Makrand Sinha. k-forrelation optimally separates quantum and classical query complexity. CoRR, abs/2008.07003, 2020. URL: http://arxiv.org/abs/2008.07003.
  4. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom generators from polarizing random walks. Theory of Computing, 15(10):1-26, 2019. URL: https://doi.org/10.4086/toc.2019.v015a010.
  5. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, and David Zuckerman. XOR lemmas for resilient functions against polynomials. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 234–246, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384242.
  6. Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.22.
  7. Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, and Avishay Tal. Improved pseudorandomness for unordered branching programs through local monotonicity. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 363-375, 2018. URL: https://doi.org/10.1145/3188745.3188800.
  8. Andreas Defant, Leonhard Frerick, Joaquim Ortega-Cerdà, Myriam Ounaïes, and Kristian Seip. The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Annals of mathematics, pages 485-497, 2011. Google Scholar
  9. Uma Girish, Ran Raz, and Wei Zhan. Lower bounds for XOR of forrelations. CoRR, abs/2007.03631, 2020. URL: http://arxiv.org/abs/2007.03631.
  10. Parikshit Gopalan, Rocco A. Servedio, Avishay Tal, and Avi Wigderson. Degree and sensitivity: tails of two distributions, 2016. URL: http://arxiv.org/abs/1604.07432.
  11. Parikshit Gopalan, Rocco A. Servedio, and Avi Wigderson. Degree and sensitivity: Tails of two distributions. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 13:1-13:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.13.
  12. Chin Ho Lee. Fourier bounds and pseudorandom generators for product tests. In Amir Shpilka, editor, 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019, New Brunswick, NJ, USA, volume 137 of LIPIcs, pages 7:1-7:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.7.
  13. Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. In 30th Annual Symposium on Foundations of Computer Science, pages 574-579. IEEE, 1989. Google Scholar
  14. Ashley Montanaro. Some applications of hypercontractive inequalities in quantum information theory. Journal of Mathematical Physics, 53(12):122206, 2012. Google Scholar
  15. Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. In Harriet Ortiz, editor, Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, 1990, Baltimore, Maryland, USA, pages 213-223. ACM, 1990. URL: https://doi.org/10.1145/100216.100244.
  16. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: https://doi.org/10.1017/CBO9781139814782.
  17. Qazi Ibadu Rahman and Gerhard Schmeisser. Analytic theory of polynomials, volume 26 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, Oxford, 2002. Google Scholar
  18. Ran Raz and Avishay Tal. Oracle separation of BQP and PH. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 13-23. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316315.
  19. Alexander A. Razborov. 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, April 1987. URL: https://doi.org/10.1007/BF01137685.
  20. Alexander A. Sherstov, Andrey A. Storozhenko, and Pei Wu. An optimal separation of randomized and quantum query complexity. Electron. Colloquium Comput. Complex., 27:128, 2020. URL: https://eccc.weizmann.ac.il/report/2020/128.
  21. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC ’87, page 77–82, New York, NY, USA, 1987. Association for Computing Machinery. URL: https://doi.org/10.1145/28395.28404.
  22. Roman Smolensky. On representations by low-degree polynomials. In Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science, SFCS ’93, page 130–138, USA, 1993. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.1993.366874.
  23. Avishay Tal. Tight bounds on the Fourier spectrum of AC0. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 15:1-15:31. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.15.
  24. Avishay Tal. Towards optimal separations between quantum and randomized query complexities. Electron. Colloquium Comput. Complex., 26:179, 2019. URL: https://eccc.weizmann.ac.il/report/2019/179.
  25. Salil P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. Google Scholar
  26. Emanuele Viola. The sum of d small-bias generators fools polynomials of degree d. Computational Complexity, 18(2):209-217, 2009. URL: https://doi.org/10.1007/s00037-009-0273-5.
  27. Emanuele Viola. Fourier conjectures, correlation bounds, and majority. Electron. Colloquium Comput. Complex., 27:175, 2020. URL: https://eccc.weizmann.ac.il/report/2020/175.
  28. Xinyu Wu. A stochastic calculus approach to the oracle separation of BQP and PH. CoRR, abs/2007.02431, 2020. URL: http://arxiv.org/abs/2007.02431.
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