Impagliazzo’s Worlds Through the Lens of Conditional Kolmogorov Complexity

Authors Zhenjian Lu , Rahul Santhanam



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Zhenjian Lu
  • University of Warwick, UK
Rahul Santhanam
  • University of Oxford, UK

Acknowledgements

We thank Shuichi Hirahara, Yanyi Liu, Igor C. Oliveira, and Hanlin Ren for useful discussions.

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Zhenjian Lu and Rahul Santhanam. Impagliazzo’s Worlds Through the Lens of Conditional Kolmogorov Complexity. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 110:1-110:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.110

Abstract

We develop new characterizations of Impagliazzo’s worlds Algorithmica, Heuristica and Pessiland by the intractability of conditional Kolmogorov complexity 𝖪 and conditional probabilistic time-bounded Kolmogorov complexity pK^t. In our first set of results, we show that NP ⊆ BPP iff pK^t(x ∣ y) can be computed efficiently in the worst case when t is sublinear in |x| + |y|; DistNP ⊆ HeurBPP iff pK^t(x ∣ y) can be computed efficiently over all polynomial-time samplable distributions when t is sublinear in |x| + |y|; and infinitely-often one-way functions fail to exist iff pK^t(x ∣ y) can be computed efficiently over all polynomial-time samplable distributions for t a sufficiently large polynomial in |x| + |y|. These results characterize Impagliazzo’s worlds Algorithmica, Heuristica and Pessiland purely in terms of the tractability of conditional pK^t. Notably, the results imply that Pessiland fails to exist iff the average-case intractability of conditional pK^t is insensitive to the difference between sublinear and polynomially bounded t. As a corollary, while we prove conditional pK^t to be NP-hard for sublinear t, showing NP-hardness for large enough polynomially bounded t would eliminate Pessiland as a possible world of average-case complexity. In our second set of results, we characterize Impagliazzo’s worlds Algorithmica, Heuristica and Pessiland by the distributional tractability of a natural problem, i.e., approximating the conditional Kolmogorov complexity, that is provably intractable in the worst case. We show that NP ⊆ BPP iff conditional Kolmogorov complexity can be approximated in the semi-worst case; and DistNP ⊆ HeurBPP iff conditional Kolmogorov complexity can be approximated on average over all independent polynomial-time samplable distributions. It follows from a result by Ilango, Ren, and Santhanam (STOC 2022) that infinitely-often one-way functions fail to exist iff conditional Kolmogorov complexity can be approximated on average over all polynomial-time samplable distributions. Together, these results yield the claimed characterizations. Our techniques, combined with previous work, also yield a characterization of auxiliary-input one-way functions and equivalences between different average-case tractability assumptions for conditional Kolmogorov complexity and its variants. Our results suggest that novel average-case tractability assumptions such as tractability in the semi-worst case and over independent polynomial-time samplable distributions might be worthy of further study.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • meta-complexity
  • Kolmogorov complexity
  • one-way functions
  • average-case complexity

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References

  1. Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-way functions and a conditional variant of MKTP. In Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 7:1-7:19, 2021. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2021.7.
  2. Harry Buhrman, Troy Lee, and Dieter van Melkebeek. Language compression and pseudorandom generators. Comput. Complex., 14(3):228-255, 2005. URL: https://doi.org/10.1007/s00037-005-0199-5.
  3. Halley Goldberg, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira. Probabilistic Kolmogorov complexity with applications to average-case complexity. In Computational Complexity Conference (CCC), pages 16:1-16:60, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.16.
  4. Johan Håstad, Russell Impagliazzo, Leonid A. Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4):1364-1396, 1999. URL: https://doi.org/10.1137/S0097539793244708.
  5. Shuichi Hirahara. Characterizing average-case complexity of PH by worst-case meta-complexity. In Symposium on Foundations of Computer Science (FOCS), pages 50-60, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00014.
  6. Shuichi Hirahara. Symmetry of information from meta-complexity. In Computational Complexity Conference (CCC), pages 26:1-26:41, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.26.
  7. Shuichi Hirahara. Capturing one-way functions via NP-hardness of meta-complexity. In Symposium on Theory of Computing (STOC), pages 1027-1038, 2023. URL: https://doi.org/10.1145/3564246.3585130.
  8. Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, and Igor C. Oliveira. A duality between one-way functions and average-case symmetry of information. In Symposium on Theory of Computing (STOC), pages 1039-1050, 2023. URL: https://doi.org/10.1145/3564246.3585138.
  9. Shuichi Hirahara and Rahul Santhanam. On the average-case complexity of MCSP and its variants. In Computational Complexity Conference (CCC), pages 7:1-7:20, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.7.
  10. Shuichi Hirahara and Rahul Santhanam. Excluding PH pessiland. In Innovations in Theoretical Computer Science Conference (ITCS), pages 85:1-85:25, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.85.
  11. Rahul Ilango, Hanlin Ren, and Rahul Santhanam. Robustness of average-case meta-complexity via pseudorandomness. In Symposium on Theory of Computing (STOC), pages 1575-1583, 2022. URL: https://doi.org/10.1145/3519935.3520051.
  12. Russell Impagliazzo. A personal view of average-case complexity. In Proceedings of the Tenth Annual Structure in Complexity Theory Conference, pages 134-147, 1995. URL: https://doi.org/10.1109/SCT.1995.514853.
  13. Russell Impagliazzo and Leonid A. Levin. No better ways to generate hard NP instances than picking uniformly at random. In Symposium on Theory of Computing (STOC), pages 812-821, 1990. URL: https://doi.org/10.1109/FSCS.1990.89604.
  14. Yanyi Liu and Rafael Pass. On one-way functions and Kolmogorov complexity. In Symposium on Foundations of Computer Science (FOCS), pages 1243-1254, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00118.
  15. Yanyi Liu and Rafael Pass. On the possibility of basing cryptography on EXP≠BPP. In International Cryptology Conference (CRYPTO), pages 11-40, 2021. URL: https://doi.org/10.1007/978-3-030-84242-0_2.
  16. Yanyi Liu and Rafael Pass. On one-way functions from NP-complete problems. In Conference on Computational Complexity (CCC), pages 36:1-36:24, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.36.
  17. Yanyi Liu and Rafael Pass. One-way functions and the hardness of (probabilistic) time-bounded Kolmogorov complexity w.r.t. samplable distributions. In Annual Cryptology Conference (CRYPTO), pages 645-673, 2023. URL: https://doi.org/10.1007/978-3-031-38545-2_21.
  18. Zhenjian Lu, Igor C. Oliveira, and Rahul Santhanam. Pseudodeterministic algorithms and the structure of probabilistic time. In Symposium on Theory of Computing (STOC), pages 303-316, 2021. URL: https://doi.org/10.1145/3406325.3451085.
  19. Zhenjian Lu, Igor C. Oliveira, and Marius Zimand. Optimal coding theorems in time-bounded Kolmogorov complexity. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 92:1-92:14, 2022. URL: https://doi.org/10.4230/LIPICS.ICALP.2022.92.
  20. Zhenjian Lu and Rahul Santhanam. Impagliazzo’s worlds through the lens of conditional Kolmogorov complexity. Electronic Colloquium on Computational Complexity (ECCC), TR24-085, 2024. URL: https://eccc.weizmann.ac.il/report/2024/085.
  21. Hanlin Ren and Rahul Santhanam. Hardness of KT characterizes parallel cryptography. In Computational Complexity Conference (CCC), pages 35:1-35:58, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.35.
  22. Michael E. Saks and Rahul Santhanam. On randomized reductions to the random strings. In Computational Complexity Conference (CCC), pages 29:1-29:30, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.29.
  23. Rahul Santhanam. Pseudorandomness and the minimum circuit size problem. In Innovations in Theoretical Computer Science Conference (ITCS), pages 68:1-68:26, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.68.
  24. Larry J. Stockmeyer. On approximation algorithms for #P. SIAM J. Comput., 14(4):849-861, 1985. URL: https://doi.org/10.1137/0214060.