Document Open Access Logo

Finding Errorless Pessiland in Error-Prone Heuristica

Authors Shuichi Hirahara, Mikito Nanashima

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2022.25.pdf
  • Filesize: 0.88 MB
  • 28 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • average-case complexity
  • oracle separation
  • relativization barrier
  • meta-complexity
  • learning
  • auxiliary-input cryptography

Metrics

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

Abstract

Average-case complexity has two standard formulations, i.e., errorless complexity and error-prone complexity. In average-case complexity, a critical topic of research is to show the equivalence between these formulations, especially on the average-case complexity of NP.
In this study, we present a relativization barrier for such an equivalence. Specifically, we construct an oracle relative to which NP is easy on average in the error-prone setting (i.e., DistNP ⊆ HeurP) but hard on average in the errorless setting even by 2^o(n/log n)-size circuits (i.e., DistNP ⊈ AvgSIZE[2^o(n/log n)]), which provides an answer to the open question posed by Impagliazzo (CCC 2011). Additionally, we show the following in the same relativized world:
- Lower bound of meta-complexity: GapMINKT^𝒪 ∉ prSIZE^𝒪[2^o(n/log n)] and GapMCSP^𝒪 ∉ prSIZE^𝒪[2^(n^ε)] for some ε > 0.
- Worst-case hardness of learning on uniform distributions: P/poly is not weakly PAC learnable with membership queries on the uniform distribution by nonuniform 2ⁿ/n^ω(1)-time algorithms.
- Average-case hardness of distribution-free learning: P/poly is not weakly PAC learnable on average by nonuniform 2^o(n/log n)-time algorithms.
- Weak cryptographic primitives: There exist a hitting set generator, an auxiliary-input one-way function, an auxiliary-input pseudorandom generator, and an auxiliary-input pseudorandom function against SIZE^𝒪[2^o(n/log n)]. 
This provides considerable insights into Pessiland (i.e., the world in which no one-way function exists, and NP is hard on average), such as the relativized separation of the error-prone average-case hardness of NP and auxiliary-input cryptography. At the core of our oracle construction is a new notion of random restriction with masks.

Cite As Get BibTex

Shuichi Hirahara and Mikito Nanashima. Finding Errorless Pessiland in Error-Prone Heuristica. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 25:1-25:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.25

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Mikito Nanashima
  • Tokyo Institute of Technology, Japan

Funding

  • Hirahara, Shuichi: JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Nanashima, Mikito: JST, ACT-X Grant Number JPMJAX190M, Japan.

Acknowledgements

The authors would like to thank the anonymous reviewers for many helpful comments.

References

  1. S Aaronson and A Wigderson. Algebrization: A New Barrier in Complexity Theory. ACM Trans. Comput. Theory, 1(1), February 2009. Google Scholar
  2. E. Allender, M. Cheraghchi, D. Myrisiotis, H. Tirumala, and I. Volkovich. One-Way Functions and a Conditional Variant of MKTP. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2021, December 15-17, 2021, Virtual Conference, volume 213 of LIPIcs, pages 7:1-7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  3. B. Applebaum, B. Barak, and D. Xiao. On Basing Lower-Bounds for Learning on Worst-Case Assumptions. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS'08, pages 211-220, 2008. Google Scholar
  4. T. Baker, J. Gill, and R. Solovay. Relativizations of the 𝒫 = ?NP Question. SIAM Journal on Computing, 4(4):431-442, 1975. Google Scholar
  5. Manuel Blum and Sampath Kannan. Designing Programs that Check Their Work. J. ACM, 42(1):269-291, 1995. URL: https://doi.org/10.1145/200836.200880.
  6. Manuel Blum and Silvio Micali. How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. SIAM J. Comput., 13(4):850-864, 1984. URL: https://doi.org/10.1137/0213053.
  7. A. Bogdanov and L. Trevisan. Average-Case Complexity. Foundations and Trends in Theoretical Computer Science, 2(1):1-106, 2006. Google Scholar
  8. G. Brassard. Relativized Cryptography. IEEE Transactions on Information Theory, 29(6):877-894, 1983. Google Scholar
  9. 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.
  10. M. Carmosino, R. Impagliazzo, V. Kabanets, and A. Kolokolova. Learning Algorithms from Natural Proofs. In Proceedings of the 31st Conference on Computational Complexity, CCC'16. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  11. R. Gennaro and L. Trevisan. Lower bounds on the efficiency of generic cryptographic constructions. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 305-313, 2000. Google Scholar
  12. O. Goldreich. Foundations of Cryptography: Volume 1. Cambridge University Press, New York, NY, USA, 2006. Google Scholar
  13. O. Goldreich, S. Goldwasser, and S. Micali. How to Construct Random Functions. J. ACM, 33(4):792-807, August 1986. Google Scholar
  14. J. Håstad. Computational Limitations of Small-depth Circuits. MIT Press, Cambridge, MA, USA, 1987. Google Scholar
  15. J. Håstad, R. Impagliazzo, L. A. Levin, and M. Luby. A Pseudorandom Generator from Any One-way Function. SIAM J. Comput., 28(4):1364-1396, March 1999. Google Scholar
  16. S. 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, 2018. Google Scholar
  17. S Hirahara. Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity. In IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pages 50-60, 2020. Google Scholar
  18. S. Hirahara. Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions. In 35th Computational Complexity Conference (CCC 2020), volume 169 of LIPIcs, pages 20:1-20:47, Dagstuhl, Germany, 2020. Google Scholar
  19. S. Hirahara. Average-Case Hardness of NP from Exponential Worst-Case Hardness Assumptions. In 53rd Annual ACM Symposium on Theory of Computing (STOC 2021), 2021. Google Scholar
  20. S. Hirahara and M. Nanashima. On Worst-Case Learning in Relativized Heuristica. In 62nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2021), 2021. Google Scholar
  21. S. Hirahara and R. Santhanam. On the Average-Case Complexity of MCSP and Its Variants. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 7:1-7:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  22. S. Hirahara and R. Santhanam. Errorless versus Error-prone Average-case Complexity. In The 13th Innovations in Theoretical Computer Science (ITCS 2022), 2022. Google Scholar
  23. R. Ilango, H. Ren, and R. Santhanam. Hardness on any Samplable Distribution Suffices: New Characterizations of One-Way Functions by Meta-Complexity. Electron. Colloquium Comput. Complex., page 82, 2021. Google Scholar
  24. R. Impagliazzo. A personal view of average-case complexity. In Proceedings of IEEE Tenth Annual Conference on Structure in Complexity Theory, pages 134-147, 1995. Google Scholar
  25. R. Impagliazzo. Relativized Separations of Worst-Case and Average-Case Complexities for NP. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 104-114, 2011. Google Scholar
  26. R. Impagliazzo and L. Levin. No better ways to generate hard NP instances than picking uniformly at random. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, FOCS'90, pages 812-821, 1990. Google Scholar
  27. R. Impagliazzo and S. Rudich. Limits on the Provable Consequences of One-Way Permutations. In Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing, STOC ’89, pages 44-61, New York, NY, USA, 1989. ACM. Google Scholar
  28. K. Ko. On the Complexity of Learning Minimum Time-Bounded Turing Machines. SIAM Journal on Computing, 20(5):962-986, 1991. Google Scholar
  29. L. A Levin. Average Case Complete Problems. SIAM J. Comput., 15(1):285-286, February 1986. Google Scholar
  30. Y. Liu and R. Pass. On One-way Functions and Kolmogorov Complexity. In IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pages 1243-1254, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00118.
  31. Y. Liu and R. Pass. A Note on One-way Functions and Sparse Languages. Electron. Colloquium Comput. Complex., page 92, 2021. Google Scholar
  32. Y. Liu and R. Pass. On the Possibility of Basing Cryptography on EXP ≠ BPP. In Advances in Cryptology - CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Virtual Event, August 16-20, 2021, Proceedings, Part I, volume 12825 of Lecture Notes in Computer Science, pages 11-40. Springer, 2021. Google Scholar
  33. M. Nanashima. A Theory of Heuristic Learnability. In Proceedings of the 34th Conference on Learning Theory, COLT'21. PMLR, August 2021. Google Scholar
  34. M. Nanashima. On Basing Auxiliary-Input Cryptography on NP-Hardness via Nonadaptive Black-Box Reductions. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of LIPIcs, pages 29:1-29:15, Dagstuhl, Germany, 2021. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. Google Scholar
  35. I. Oliveira and R. Santhanam. Conspiracies between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness. In Proceedings of the 32nd Computational Complexity Conference, CCC'17, Dagstuhl, DEU, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  36. R. Ostrovsky. One-way functions, hard on average problems, and statistical zero-knowledge proofs. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference, pages 133-138, 1991. Google Scholar
  37. R. Ostrovsky and A. Wigderson. One-way functions are essential for non-trivial zero-knowledge. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems, ISTCS'93, pages 3-17, June 1993. Google Scholar
  38. A. A. Razborov and S. Rudich. Natural Proofs. Journal of Computer and System Sciences, 55(1):24-35, 1997. Google Scholar
  39. H. Ren and R. Santhanam. A Relativization Perspective on Meta-Complexity. In The 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), 2022. Google Scholar
  40. L. Valiant. A Theory of the Learnable. Commun. ACM, 27(11):1134-1142, 1984. URL: https://doi.org/10.1145/1968.1972.
  41. Thomas Watson. Relativized Worlds without Worst-Case to Average-Case Reductions for NP. ACM Trans. Comput. Theory, 4(3), September 2012. Google Scholar
  42. H Wee. Finding Pessiland. In Proceedings of the Third Conference on Theory of Cryptography, TCC'06, pages 429-442, Berlin, Heidelberg, 2006. Springer-Verlag. Google Scholar
  43. D. Xiao. On basing ZK ≠ BPP on the hardness of PAC learning. In Proceedings of the 24th Conference on Computational Complexity, CCC'09, pages 304-315, 2009. Google Scholar
  44. A. Yao. Theory and Application of Trapdoor Functions. In Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, FOCS'82, pages 80-91, November 1982. Google Scholar
  45. M. Zimand. Efficient Privatization of Random Bits. In IN WORKSHOP ON RANDOMIZED ALGORITHMS, 1997. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact