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The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False

Authors Noam Mazor, Rafael Pass

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Author Details

Noam Mazor
  • Cornell Tech, New York, NY, USA
Rafael Pass
  • Tel Aviv University, Israel
  • Cornell Tech, New York, NY, USA

Funding

  • Mazor, Noam: Research partly supported by NSF CNS-2149305 and NSF CNS-2128519.
  • Pass, Rafael: Supported in part by NSF Award CNS 2149305, AFOSR Award FA9550-18-1-0267, AFOSR Award FA9550-23-1-0387, a JP Morgan Faculty Award and an Algorand Foundation award. This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government, DARPA, AFOSR or the Algorand Foundation.

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Noam Mazor and Rafael Pass. The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.80

Abstract

The Perebor (Russian for "brute-force search") conjectures, which date back to the 1950s and 1960s are some of the oldest conjectures in complexity theory. The conjectures are a stronger form of the NP ≠ P conjecture (which they predate) and state that for "meta-complexity" problems, such as the Time-bounded Kolmogorov complexity Problem, and the Minimum Circuit Size Problem, there are no better algorithms than brute force search. In this paper, we disprove the non-uniform version of the Perebor conjecture for the Time-Bounded Kolmogorov complexity problem. We demonstrate that for every polynomial t(⋅), there exists of a circuit of size 2^{4n/5+o(n)} that solves the t(⋅)-bounded Kolmogorov complexity problem on every instance. Our algorithm is black-box in the description of the Universal Turing Machine U employed in the definition of Kolmogorov Complexity and leverages the characterization of one-way functions through the hardness of the time-bounded Kolmogorov complexity problem of Liu and Pass (FOCS'20), and the time-space trade-off for one-way functions of Fiat and Naor (STOC'91). We additionally demonstrate that no such black-box algorithm can have circuit size smaller than 2^{n/2-o(n)}. Along the way (and of independent interest), we extend the result of Fiat and Naor and demonstrate that any efficiently computable function can be inverted (with probability 1) by a circuit of size 2^{4n/5+o(n)}; as far as we know, this yields the first formal proof that a non-trivial circuit can invert any efficient function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Kolmogorov complexity
  • perebor conjecture
  • function inversion

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References

  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. An 0 (n log n) sorting network. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 1-9, 1983. Google Scholar
  2. Mohmammad Bavarian. Lecture 6. Lecture notes in "6.S897: Algebra and Computation" by Madhu Sudan, 2012. Google Scholar
  3. Gregory J. Chaitin. On the simplicity and speed of programs for computing infinite sets of natural numbers. J. ACM, 16(3):407-422, 1969. Google Scholar
  4. Kai-Min Chung, Edward Lui, Mohammad Mahmoody, and Rafael Pass. Unprovable security of 2-message zero knowledge. Cryptology ePrint Archive, 2012. Google Scholar
  5. Sandro Coretti, Yevgeniy Dodis, Siyao Guo, and John Steinberger. Random oracles and non-uniformity. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 227-258. Springer, 2018. Google Scholar
  6. Whitfield Diffie and Martin E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, pages 644-654, 1976. Google Scholar
  7. Amos Fiat and Moni Naor. Rigorous time/space trade-offs for inverting functions. SIAM Journal on Computing, 29(3):790-803, 2000. Google Scholar
  8. Gudmund Skovbjerg Frandsen and Peter Bro Miltersen. Reviewing bounds on the circuit size of the hardest functions. Information processing letters, 95(2):354-357, 2005. Google Scholar
  9. Rosario Gennaro, Yael Gertner, Jonathan Katz, and Luca Trevisan. Bounds on the efficiency of generic cryptographic constructions. SIAM Journal on Computing, 35(1):217-246, 2005. URL: http://epubs.siam.org/SICOMP/VOLUME-35/art_44327.html.
  10. J. Hartmanis. Generalized kolmogorov complexity and the structure of feasible computations. In 24th Annual Symposium on Foundations of Computer Science (sfcs 1983), pages 439-445, November 1983. URL: https://doi.org/10.1109/SFCS.1983.21.
  11. Martin Hellman. A cryptanalytic time-memory trade-off. IEEE transactions on Information Theory, 26(4):401-406, 1980. Google Scholar
  12. Russell Impagliazzo. Relativized separations of worst-case and average-case complexities for np. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 104-114. IEEE, 2011. Google Scholar
  13. Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 73-79, 2000. Google Scholar
  14. Ker-I Ko. On the notion of infinite pseudorandom sequences. Theor. Comput. Sci., 48(3):9-33, 1986. URL: https://doi.org/10.1016/0304-3975(86)90081-2.
  15. A. N. Kolmogorov. Three approaches to the quantitative definition of information. International Journal of Computer Mathematics, 2(1-4):157-168, 1968. Google Scholar
  16. Yanyi Liu and Rafael Pass. On one-way functions and kolmogorov complexity. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1243-1254. IEEE, 2020. Google Scholar
  17. Yanyi Liu and Rafael Pass. On one-way functions and the worst-case hardness of time-bounded kolmogorov complexity. Cryptology ePrint Archive, 2023. Google Scholar
  18. Noam Mazor and Rafael Pass. The non-uniform perebor conjecture for time-bounded kolmogorov complexity is false. Technical Report TR23-175, Electronic Colloquium on Computational Complexity, 2023. Google Scholar
  19. Hanlin Ren and Rahul Santhanam. Hardness of kt characterizes parallel cryptography. In 36th Computational Complexity Conference (CCC 2021). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  20. Michael Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC), pages 330-335, 1983. Google Scholar
  21. R.J. Solomonoff. A formal theory of inductive inference. part i. Information and Control, 7(1):1-22, 1964. URL: https://doi.org/10.1016/S0019-9958(64)90223-2.
  22. Boris A Trakhtenbrot. A survey of russian approaches to perebor (brute-force searches) algorithms. Annals of the History of Computing, 6(4):384-400, 1984. Google Scholar
  23. Sergey Yablonski. The algorithmic difficulties of synthesizing minimal switching circuits. Problemy Kibernetiki, 2(1):75-121, 1959. Google Scholar
  24. Sergey V Yablonski. On the impossibility of eliminating perebor in solving some problems of circuit theory. Doklady Akademii Nauk SSSR, 124(1):44-47, 1959. Google Scholar
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