Document Open Access Logo

Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

Authors William M. Hoza , Edward Pyne, Salil Vadhan



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2021.7.pdf
  • Filesize: 0.6 MB
  • 20 pages

Document Identifiers

Author Details

William M. Hoza
  • University of Texas at Austin, TX, USA
Edward Pyne
  • Harvard University, Cambridge, MA, USA
Salil Vadhan
  • Harvard University, Cambridge, MA, USA

Acknowledgements

We thank Jack Murtagh for collaboration at the start of this research. The first author thanks Dean Doron for insightful and relevant discussions about the works by Murtagh et al. [Murtagh et al., 2017; Murtagh et al., 2019]. We thank Shyam Narayanan, Dean Doron and David Zuckerman for valuable comments on a draft of this paper.

Cite AsGet BibTex

William M. Hoza, Edward Pyne, and Salil Vadhan. Pseudorandom Generators for Unbounded-Width Permutation Branching Programs. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 7:1-7:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.7

Abstract

We prove that the Impagliazzo-Nisan-Wigderson [Impagliazzo et al., 1994] pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of Õ(log d + log n ⋅ log(1/ε)), assuming the program has only one accepting vertex in the final layer. Here, n is the length of the program, d is the degree (equivalently, the alphabet size), and ε is the error of the PRG. In contrast, we show that a randomly chosen generator requires seed length Ω(n log d) to fool such unbounded-width programs. Thus, this is an unusual case where an explicit construction is "better than random." Except when the program’s width w is very small, this is an improvement over prior work. For example, when w = poly(n) and d = 2, the best prior PRG for permutation branching programs was simply Nisan’s PRG [Nisan, 1992], which fools general ordered branching programs with seed length O(log(wn/ε) log n). We prove a seed length lower bound of Ω̃(log d + log n ⋅ log(1/ε)) for fooling these unbounded-width programs, showing that our seed length is near-optimal. In fact, when ε ≤ 1/log n, our seed length is within a constant factor of optimal. Our analysis of the INW generator uses the connection between the PRG and the derandomized square of Rozenman and Vadhan [Rozenman and Vadhan, 2005] and the recent analysis of the latter in terms of unit-circle approximation by Ahmadinejad et al. [Ahmadinejad et al., 2020].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandom generators
  • permutation branching programs

Metrics

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

References

  1. AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, and Salil Vadhan. High-precision estimation of random walks in small space. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020. To appear. Google Scholar
  2. Mark Braverman, Anup Rao, Ran Raz, and Amir Yehudayoff. Pseudorandom generators for regular branching programs. SIAM Journal on Computing, 43(3):973-986, 2014. URL: https://doi.org/10.1137/120875673.
  3. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom generators from polarizing random walks. Theory of Computing. An Open Access Journal, 15:Paper No. 10, 26, 2019. URL: https://doi.org/10.4086/toc.2019.v015a010.
  4. Michael B. Cohen, Jonathan Kelner, John Peebles, Richard Peng, Anup B. Rao, Aaron Sidford, and Adrian Vladu. Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 410-419. ACM, New York, 2017. URL: https://doi.org/10.1145/3055399.3055463.
  5. Anindya De. Pseudorandomness for permutation and regular branching programs. In 26th Annual IEEE Conference on Computational Complexity, pages 221-231. IEEE Computer Soc., Los Alamitos, CA, 2011. URL: https://doi.org/10.1109/CCC.2011.23.
  6. Anindya De, Omid Etesami, Luca Trevisan, and Madhur Tulsiani. Improved pseudorandom generators for depth 2 circuits. In Approximation, randomization, and combinatorial optimization, volume 6302 of Lecture Notes in Computer Science, pages 504-517. Springer, Berlin, 2010. URL: https://doi.org/10.1007/978-3-642-15369-3_38.
  7. Yevgeniy Dodis, Rafail Ostrovsky, Leonid Reyzin, and Adam Smith. Fuzzy extractors: how to generate strong keys from biometrics and other noisy data. SIAM Journal on Computing, 38(1):97-139, 2008. URL: https://doi.org/10.1137/060651380.
  8. Shlomo Hoory and Avi Wigderson. Universal traversal sequences for expander graphs. Information Processing Letters, 46(2):67-69, 1993. URL: https://doi.org/10.1016/0020-0190(93)90199-J.
  9. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC), page 356–364, New York, NY, USA, 1994. Association for Computing Machinery. URL: https://doi.org/10.1145/195058.195190.
  10. Michal Koucký, Prajakta Nimbhorkar, and Pavel Pudlák. Pseudorandom generators for group products. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pages 263-272. ACM, New York, 2011. URL: https://doi.org/10.1145/1993636.1993672.
  11. M. Luby and B. Veličković. On deterministic approximation of DNF. Algorithmica, 16(4-5):415-433, 1996. URL: https://doi.org/10.1007/s004539900054.
  12. Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. SIAM Journal on Computing, 42(3):1275-1301, 2013. URL: https://doi.org/10.1137/100811623.
  13. Jack Murtagh, Omer Reingold, Aaron Sidford, and Salil Vadhan. Derandomization beyond connectivity: undirected Laplacian systems in nearly logarithmic space. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 801-812. IEEE Computer Soc., Los Alamitos, CA, 2017. URL: https://doi.org/10.1109/FOCS.2017.79.
  14. Jack Murtagh, Omer Reingold, Aaron Sidford, and Salil Vadhan. Deterministic Approximation of Random Walks in Small Space. In Dimitris Achlioptas and László A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 42:1-42:22, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.42.
  15. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. URL: https://doi.org/10.1007/BF01305237.
  16. Omer Reingold. Undirected connectivity in log-space. Journal of the ACM, 55(4):Art. 17, 24, 2008. URL: https://doi.org/10.1145/1391289.1391291.
  17. Omer Reingold, Thomas Steinke, and Salil Vadhan. Pseudorandomness for regular branching programs via Fourier analysis. In Approximation, randomization, and combinatorial optimization, volume 8096 of Lecture Notes in Computer Science, pages 655-670. Springer, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_45.
  18. Omer Reingold, Luca Trevisan, and Salil Vadhan. Pseudorandom walks on regular digraphs and the RL vs. L problem. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 457-466. ACM, New York, 2006. URL: https://doi.org/10.1145/1132516.1132583.
  19. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics. Second Series, 155(1):157-187, 2002. URL: https://doi.org/10.2307/3062153.
  20. Eyal Rozenman and Salil Vadhan. Derandomized squaring of graphs. In Approximation, randomization and combinatorial optimization, volume 3624 of Lecture Notes in Computer Science, pages 436-447. Springer, Berlin, 2005. URL: https://doi.org/10.1007/11538462_37.
  21. Alexander Schrijver. Combinatorial optimization. Polyhedra and efficiency. Vol. A, volume 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 2003. Paths, flows, matchings, Chapters 1-38. Google Scholar
  22. Thomas Steinke. Pseudorandomness for permutation branching programs without the group theory. ECCC preprint TR12-083, 2012. 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