Document Open Access Logo

Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions

Authors Lijie Chen, Shuichi Hirahara, Neekon Vafa

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2022.45.pdf
  • Filesize: 0.79 MB
  • 16 pages

Document Identifiers

Author Details

Lijie Chen
  • MIT, Boston, MA, USA
Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Neekon Vafa
  • MIT, Boston, MA, USA

Funding

  • Chen, Lijie: supported by NSF CCF-2127597 and an IBM Fellowship.
  • Hirahara, Shuichi: supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Vafa, Neekon: supported by NSF fellowship DGE-1745302.

Acknowledgements

We are grateful to Rahul Ilango and Ryan Williams for helpful discussions. In particular, we want to thank Ryan Williams for the observation that an HSG with seed length O(log t) already suffices for our proof.

Cite As Get BibTex

Lijie Chen, Shuichi Hirahara, and Neekon Vafa. Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.45

Abstract

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness of NP or PH:  
- NTIME[n] cannot be solved in quasi-linear time on average if UP ̸ ⊆ DTIME[2^{Õ(√n)}].
- Σ₂TIME[n] cannot be solved in quasi-linear time on average if Σ_kSAT cannot be solved in time 2^{Õ(√n)} for some constant k. Previously, it was not known if even average-case hardness of Σ₃SAT implies the average-case hardness of Σ₂TIME[n].
- Under the Exponential-Time Hypothesis (ETH), there is no average-case n^{1+ε}-time algorithm for NTIME[n] whose running time can be estimated in time n^{1+ε} for some constant ε > 0. 
Our results are given by generalizing the non-black-box worst-case-to-average-case connections presented by Hirahara (STOC 2021) to the settings of fine-grained complexity. To do so, we construct quite efficient complexity-theoretic pseudorandom generators under the assumption that the nondeterministic linear time is easy on average, which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • Average-case complexity
  • worst-case to average-case reduction

Metrics

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

Related Versions

References

  1. Adi Akavia, Oded Goldreich, Shafi Goldwasser, and Dana Moshkovitz. On basing one-way functions on NP-hardness. In Proceedings of the Symposium on Theory of Computing (STOC), pages 701-710, 2006. URL: https://doi.org/10.1145/1132516.1132614.
  2. Luis Antunes, Lance Fortnow, Dieter van Melkebeek, and N. V. Vinodchandran. Computational depth: Concept and applications. Theor. Comput. Sci., 354(3):391-404, 2006. URL: https://doi.org/10.1016/j.tcs.2005.11.033.
  3. Luis Filipe Coelho Antunes and Lance Fortnow. Worst-case running times for average-case algorithms. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15-18 July 2009, pages 298-303. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/CCC.2009.12.
  4. Marshall Ball, Alon Rosen, Manuel Sabin, and Prashant Nalini Vasudevan. Average-case fine-grained hardness. In Proceedings of the Symposium on Theory of Computing (STOC), pages 483-496, 2017. URL: https://doi.org/10.1145/3055399.3055466.
  5. Marshall Ball, Alon Rosen, Manuel Sabin, and Prashant Nalini Vasudevan. Proofs of Work From Worst-Case Assumptions. In Advances in Cryptology - CRYPTO 2018 - 38th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2018, Proceedings, Part I, pages 789-819, 2018. URL: https://doi.org/10.1007/978-3-319-96884-1_26.
  6. Donald Beaver, Joan Feigenbaum, Joe Kilian, and Phillip Rogaway. Locally Random Reductions: Improvements and Applications. J. Cryptol., 10(1):17-36, 1997. URL: https://doi.org/10.1007/s001459900017.
  7. Shai Ben-David, Benny Chor, Oded Goldreich, and Michael Luby. On the Theory of Average Case Complexity. J. Comput. Syst. Sci., 44(2):193-219, 1992. URL: https://doi.org/10.1016/0022-0000(92)90019-F.
  8. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. Short pcps verifiable in polylogarithmic time. In 20th Annual IEEE Conference on Computational Complexity (CCC 2005), 11-15 June 2005, San Jose, CA, USA, pages 120-134. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/CCC.2005.27.
  9. Eli Ben-Sasson and Emanuele Viola. Short PCPs with projection queries. In Proc. 41st Internat. Colloq. on Automata, Languages and Programming (ICALP'14), pages 163-173. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_14.
  10. Andrej Bogdanov and Christina Brzuska. On Basing Size-Verifiable One-Way Functions on NP-Hardness. In Proceedings of the Theory of Cryptography Conference (TCC), pages 1-6, 2015. URL: https://doi.org/10.1007/978-3-662-46494-6_1.
  11. Andrej Bogdanov and Luca Trevisan. Average-Case Complexity. Foundations and Trends in Theoretical Computer Science, 2(1), 2006. URL: https://doi.org/10.1561/0400000004.
  12. Andrej Bogdanov and Luca Trevisan. On Worst-Case to Average-Case Reductions for NP Problems. SIAM J. Comput., 36(4):1119-1159, 2006. URL: https://doi.org/10.1137/S0097539705446974.
  13. Enric Boix-Adserà, Matthew S. Brennan, and Guy Bresler. The Average-Case Complexity of Counting Cliques in Erdős-Rényi Hypergraphs. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1256-1280, 2019. URL: https://doi.org/10.1109/FOCS.2019.00078.
  14. Zvika Brakerski, Noah Stephens-Davidowitz, and Vinod Vaikuntanathan. On the hardness of average-case k-sum. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pages 29:1-29:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.29.
  15. Harry Buhrman, Lance Fortnow, and Aduri Pavan. Some results on derandomization. Theory Comput. Syst., 38(2):211-227, 2005. URL: https://doi.org/10.1007/s00224-004-1194-y.
  16. Lijie Chen, Xin Lyu, and R. Ryan Williams. Almost-everywhere circuit lower bounds from non-trivial derandomization. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1-12. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00009.
  17. Lijie Chen and Roei Tell. Simple and fast derandomization from very hard functions: eliminating randomness at almost no cost. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 283-291. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451059.
  18. Mina Dalirrooyfard, Andrea Lincoln, and Virginia Vassilevska Williams. New Techniques for Proving Fine-Grained Average-Case Hardness. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 774-785, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00077.
  19. Joan Feigenbaum and Lance Fortnow. Random-Self-Reducibility of Complete Sets. SIAM J. Comput., 22(5):994-1005, 1993. URL: https://doi.org/10.1137/0222061.
  20. Oded Goldreich and Guy N. Rothblum. Counting t-Cliques: Worst-Case to Average-Case Reductions and Direct Interactive Proof Systems. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 77-88, 2018. URL: https://doi.org/10.1109/FOCS.2018.00017.
  21. Joachim Grollmann and Alan L. Selman. Complexity Measures for Public-Key Cryptosystems. SIAM J. Comput., 17(2):309-335, 1988. URL: https://doi.org/10.1137/0217018.
  22. Yuri Gurevich and Saharon Shelah. Expected Computation Time for Hamiltonian Path Problem. SIAM J. Comput., 16(3):486-502, 1987. URL: https://doi.org/10.1137/0216034.
  23. Venkatesan Guruswami, Daniele Micciancio, and Oded Regev. The complexity of the covering radius problem. Comput. Complex., 14(2):90-121, 2005. URL: https://doi.org/10.1007/s00037-005-0193-y.
  24. Juris Hartmanis and Richard E Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117:285-306, 1965. Google Scholar
  25. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within NP. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 247-258. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00032.
  26. Shuichi Hirahara. Characterizing average-case complexity of PH by worst-case meta-complexity. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 50-60. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00014.
  27. Shuichi Hirahara. Non-disjoint promise problems from meta-computational view of pseudorandom generator constructions. In 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 20:1-20:47. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.20.
  28. Shuichi Hirahara. Average-case hardness of NP from exponential worst-case hardness assumptions. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 292-302. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451065.
  29. Shuichi Hirahara and Mikito Nanashima. On worst-case learning in relativized heuristica. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS, 2021. Google Scholar
  30. Shuichi Hirahara and Nobutaka Shimizu. Nearly Optimal Average-Case Complexity of Counting Bicliques Under SETH. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2346-2365, 2021. URL: https://doi.org/10.1137/1.9781611976465.140.
  31. Russell Impagliazzo. A Personal View of Average-Case Complexity. In Proceedings of the Structure in Complexity Theory Conference, pages 134-147, 1995. URL: https://doi.org/10.1109/SCT.1995.514853.
  32. Russell Impagliazzo. Relativized Separations of Worst-Case and Average-Case Complexities for NP. In Proceedings of the Conference on Computational Complexity (CCC), pages 104-114, 2011. URL: https://doi.org/10.1109/CCC.2011.34.
  33. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: exponential time vs. probabilistic polynomial time. J. Comput. Syst. Sci., 65(4):672-694, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00024-7.
  34. Russell Impagliazzo and Michael Luby. One-way Functions are Essential for Complexity Based Cryptography (Extended Abstract). In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 230-235, 1989. URL: https://doi.org/10.1109/SFCS.1989.63483.
  35. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  36. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which Problems Have Strongly Exponential Complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  37. Hamid Jahanjou, Eric Miles, and Emanuele Viola. Local Reductions. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 749-760, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_61.
  38. Ker-I Ko. On Some Natural Complete Operators. Theor. Comput. Sci., 37:1-30, 1985. URL: https://doi.org/10.1016/0304-3975(85)90085-4.
  39. Rio LaVigne, Andrea Lincoln, and Virginia Vassilevska Williams. Public-Key Cryptography in the Fine-Grained Setting. IACR Cryptology ePrint Archive, 2019:625, 2019. URL: https://eprint.iacr.org/2019/625.
  40. Noam Livne. On the Construction of One-Way Functions from Average Case Hardness. In Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 5-7, 2010. Proceedings, pages 301-309, 2010. URL: http://conference.iiis.tsinghua.edu.cn/ICS2010/content/papers/24.html.
  41. Cody D. Murray and R. Ryan Williams. Circuit lower bounds for nondeterministic quasi-polytime from a new easy witness lemma. SIAM J. Comput., 49(5), 2020. URL: https://doi.org/10.1137/18M1195887.
  42. Rahul Santhanam and Richard Ryan Williams. Beating Exhaustive Search for Quantified Boolean Formulas and Connections to Circuit Complexity. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 231-241, 2015. URL: https://doi.org/10.1137/1.9781611973730.18.
  43. Ronen Shaltiel and Christopher Umans. Simple extractors for all min-entropies and a new pseudorandom generator. J. ACM, 52(2):172-216, 2005. URL: https://doi.org/10.1145/1059513.1059516.
  44. Emanuele Viola. On Constructing Parallel Pseudorandom Generators from One-Way Functions. In Proceedings of the Conference on Computational Complexity (CCC), pages 183-197, 2005. URL: https://doi.org/10.1109/CCC.2005.16.
  45. Emanuele Viola. The complexity of constructing pseudorandom generators from hard functions. Computational Complexity, 13(3-4):147-188, 2005. URL: https://doi.org/10.1007/s00037-004-0187-1.
  46. Hoeteck Wee. Finding Pessiland. In Proceedings of the Theory of Cryptography Conference (TCC), pages 429-442, 2006. URL: https://doi.org/10.1007/11681878_22.
  47. R. Ryan Williams. Natural proofs versus derandomization. SIAM J. Comput., 45(2):497-529, 2016. URL: https://doi.org/10.1137/130938219.
  48. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.023.
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