Document Open Access Logo

Gap MCSP Is Not (Levin) NP-Complete in Obfustopia

Authors Noam Mazor, Rafael Pass

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2024.36.pdf
  • Filesize: 0.8 MB
  • 21 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Kolmogorov complexity
  • MCSP
  • Levin Reduction

Metrics

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

Abstract

We demonstrate that under believable cryptographic hardness assumptions, Gap versions of standard meta-complexity problems, such as the Minimum Circuit Size Problem (MCSP) and the Minimum Time-Bounded Kolmogorov Complexity problem (MKTP) are not NP-complete w.r.t. Levin (i.e., witness-preserving many-to-one) reductions. In more detail:  
- Assuming the existence of indistinguishability obfuscation, and subexponentially-secure one-way functions, an appropriate Gap version of MCSP is not NP-complete under randomized Levin-reductions. 
- Assuming the existence of subexponentially-secure indistinguishability obfuscation, subexponentially-secure one-way functions and injective PRGs, an appropriate Gap version of MKTP is not NP-complete under randomized Levin-reductions.

Cite As Get BibTex

Noam Mazor and Rafael Pass. Gap MCSP Is Not (Levin) NP-Complete in Obfustopia. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.36

Author Details

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

Funding

  • Mazor, Noam: Research partly supported by NSF CNS-2149305 and DARPA under Agreement No. HR00110C0086.
  • Pass, Rafael: Supported in part by AFOSR Award FA9550-23-1-0387, AFOSR Award FA9550-23-1-0312, and an Algorand Foundation grant. 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.

References

  1. Shweta Agrawal. Indistinguishability obfuscation without multilinear maps: new methods for bootstrapping and instantiation. In Advances in Cryptology-EUROCRYPT 2019: 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Darmstadt, Germany, May 19-23, 2019, Proceedings, Part I 38, pages 191-225. Springer, 2019. Google Scholar
  2. Shweta Agrawal and Alice Pellet-Mary. Indistinguishability obfuscation without maps: Attacks and fixes for noisy linear fe. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 110-140. Springer, 2020. Google Scholar
  3. Eric Allender and Shuichi Hirahara. New insights on the (non-) hardness of circuit minimization and related problems. ACM Transactions on Computation Theory (ToCT), 11(4):1-27, 2019. Google Scholar
  4. Eric Allender, Michal Kouckỳ, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences, 77(1):14-40, 2011. Google Scholar
  5. Prabhanjan Ananth, Aayush Jain, Huijia Lin, Christian Matt, and Amit Sahai. Indistinguishability obfuscation without multilinear maps: new paradigms via low degree weak pseudorandomness and security amplification. In Annual International Cryptology Conference, pages 284-332. Springer, 2019. Google Scholar
  6. Prabhanjan Ananth, Aayush Jain, and Amit Sahai. Indistinguishability obfuscation without multilinear maps: io from lwe, bilinear maps, and weak pseudorandomness. Cryptology ePrint Archive, 2018. Google Scholar
  7. Prabhanjan Ananth and Abhishek Jain. Indistinguishability obfuscation from compact functional encryption. In Annual Cryptology Conference, pages 308-326. Springer, 2015. Google Scholar
  8. Prabhanjan Ananth, Abhishek Jain, and Amit Sahai. Indistinguishability obfuscation for turing machines: constant overhead and amortization. In Annual International Cryptology Conference, pages 252-279. Springer, 2017. Google Scholar
  9. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM (JACM), 45(3):501-555, 1998. Google Scholar
  10. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of np. Journal of the ACM (JACM), 45(1):70-122, 1998. Google Scholar
  11. Boaz Barak and Oded Goldreich. Universal arguments and their applications. SIAM Journal on Computing, 38(5):1661-1694, 2009. Google Scholar
  12. Boaz Barak, Oded Goldreich, Rusell Impagliazzo, Steven Rudich, Amit Sahai, Salil Vadhan, and Ke Yang. On the (im) possibility of obfuscating programs. In Annual international cryptology conference, pages 1-18. Springer, 2001. Google Scholar
  13. Elette Boyle, Kai-Min Chung, and Rafael Pass. On extractability obfuscation. In Theory of cryptography conference, pages 52-73. Springer, 2014. Google Scholar
  14. Zvika Brakerski, Nico Döttling, Sanjam Garg, and Giulio Malavolta. Factoring and pairings are not necessary for io: Circular-secure lwe suffices. Cryptology ePrint Archive, 2020. Google Scholar
  15. Zvika Brakerski, Nico Döttling, Sanjam Garg, and Giulio Malavolta. Candidate iO from homomorphic encryption schemes. Journal of Cryptology, 36(3):27, 2023. Google Scholar
  16. 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
  17. Stephen A. Cook. The complexity of theorem-proving procedures. In Annual ACM Symposium on Theory of Computing (STOC), pages 151-158, 1971. Google Scholar
  18. Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. A new multilayered pcp and the hardness of hypergraph vertex cover. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 595-601, 2003. Google Scholar
  19. Uriel Fiege, Amos Fiat, and Adi Shamir. Zero knowledge proofs of identity. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 210-217, 1987. Google Scholar
  20. Sanjam Garg, Craig Gentry, Shai Halevi, Mariana Raykova, Amit Sahai, and Brent Waters. Candidate indistinguishability obfuscation and functional encryption for all circuits. SIAM Journal on Computing, 45(3):882-929, 2016. Google Scholar
  21. Romain Gay, Aayush Jain, Huijia Lin, and Amit Sahai. Indistinguishability obfuscation from simple-to-state hard problems: New assumptions, new techniques, and simplification. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 97-126. Springer, 2021. Google Scholar
  22. Romain Gay and Rafael Pass. Indistinguishability obfuscation from circular security. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 736-749, 2021. Google Scholar
  23. Craig Gentry, Allison Bishop Lewko, Amit Sahai, and Brent Waters. Indistinguishability obfuscation from the multilinear subgroup elimination assumption. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 151-170. IEEE, 2015. Google Scholar
  24. Oded Goldreich. Computational complexity: A conceptual perspective, 2008. Google Scholar
  25. Shafi Goldwasser and Guy N Rothblum. On best-possible obfuscation. Journal of Cryptology, 27(3):480-505, 2014. Google Scholar
  26. 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, 1983. URL: https://doi.org/10.1109/SFCS.1983.21.
  27. Shuichi Hirahara. NP-hardness of learning programs and partial mcsp. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 968-979. IEEE, 2022. Google Scholar
  28. Shuichi Hirahara. Symmetry of information from meta-complexity. In 37th Computational Complexity Conference (CCC 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  29. Yizhi Huang, Rahul Ilango, and Hanlin Ren. NP-hardness of approximating meta-complexity: A cryptographic approach. Cryptology ePrint Archive, 2023. Google Scholar
  30. Rahul Ilango. Approaching MCSP from above and below: Hardness for a conditional variant and AC⁰[p]. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  31. Rahul Ilango. SAT reduces to the minimum circuit size problem with a random oracle. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 733-742. IEEE, 2023. Google Scholar
  32. Rahul Ilango, Bruno Loff, and Igor Carboni Oliveira. NP-hardness of circuit minimization for multi-output functions. In CCC'20: Proceedings of the 35th Computational Complexity Conference, pages 1-36, 2020. Google Scholar
  33. Russell Impagliazzo, Valentine Kabanets, and Ilya Volkovich. The power of natural properties as oracles. computational complexity, 32(2):6, 2023. Google Scholar
  34. Russell Impagliazzo, Valentine Kabanets, and Ilya Volkovich. Synergy between circuit obfuscation and circuit minimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  35. Aayush Jain, Huijia Lin, Christian Matt, and Amit Sahai. How to leverage hardness of constant-degree expanding polynomials over ℝ to build i𝒪. In Advances in Cryptology-EUROCRYPT 2019: 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Darmstadt, Germany, May 19-23, 2019, Proceedings, Part I 38, pages 251-281. Springer, 2019. Google Scholar
  36. Aayush Jain, Huijia Lin, and Amit Sahai. Indistinguishability obfuscation from well-founded assumptions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 60-73, 2021. Google Scholar
  37. 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
  38. Richard M. Karp. Reducibility among combinatorial problems. In J. W. Thatcher and R. E. Miller, editors, Complexity of Computer Computations, pages 85-103. Plenum Press, Inc., 1972. Google Scholar
  39. 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.
  40. Ker-I Ko. On the complexity of learning minimum time-bounded turing machines. SIAM Journal on Computing, 20(5):962-986, 1991. Google Scholar
  41. A. N. Kolmogorov. Three approaches to the quantitative definition of information. International Journal of Computer Mathematics, 2(1-4):157-168, 1968. Google Scholar
  42. Ilan Komargodski, Tal Moran, Moni Naor, Rafael Pass, Alon Rosen, and Eylon Yogev. One-way functions and (im) perfect obfuscation. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 374-383. IEEE, 2014. Google Scholar
  43. Venkata Koppula, Allison Bishop Lewko, and Brent Waters. Indistinguishability obfuscation for turing machines with unbounded memory. In Proceedings of the forty-seventh annual ACM symposium on Theory of Computing, pages 419-428, 2015. Google Scholar
  44. Leonid A. Levin. Universal'nyĭe perebornyĭezadachi (Universal search problems : in Russian). Problemy Peredachi Informatsii, pages 265-266, 1973. Google Scholar
  45. Leonid Anatolevich Levin. Universal sequential search problems. Problemy peredachi informatsii, 9(3):115-116, 1973. Google Scholar
  46. Huijia Lin. Indistinguishability obfuscation from constant-degree graded encoding schemes. In Advances in Cryptology - EUROCRYPT 2016: 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, Austria, May 8-12, 2016, Proceedings, Part I 35, pages 28-57. Springer, 2016. Google Scholar
  47. Huijia Lin. Indistinguishability obfuscation from sxdh on 5-linear maps and locality-5 prgs. In Annual International Cryptology Conference, pages 599-629. Springer, 2017. Google Scholar
  48. Huijia Lin, Rafael Pass, Karn Seth, and Sidharth Telang. Output-compressing randomized encodings and applications. In Theory of Cryptography Conference, pages 96-124. Springer, 2015. Google Scholar
  49. Huijia Lin and Stefano Tessaro. Indistinguishability obfuscation from trilinear maps and block-wise local prgs. In Annual International Cryptology Conference, pages 630-660. Springer, 2017. Google Scholar
  50. Huijia Lin and Vinod Vaikuntanathan. Indistinguishability obfuscation from ddh-like assumptions on constant-degree graded encodings. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 11-20. IEEE, 2016. Google Scholar
  51. Yanyi Liu and Rafael Pass. On one-way functions from NP-complete problems. In 37th Computational Complexity Conference, 2022. Google Scholar
  52. Noam Mazor and Rafael Pass. Gap MCSP is not (levin) NP-complete in obfustopia. Cryptology ePrint Archive, 2024. Google Scholar
  53. Cody D Murray and R Ryan Williams. On the (non) NP-hardness of computing circuit complexity. Theory of Computing, 13(1):1-22, 2017. Google Scholar
  54. Moni Naor and Moti Yung. Universal one-way hash functions and their cryptographic applications. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC), pages 33-43, 1989. Google Scholar
  55. Rafael Pass, Karn Seth, and Sidharth Telang. Indistinguishability obfuscation from semantically-secure multilinear encodings. In Advances in Cryptology-CRYPTO 2014: 34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17-21, 2014, Proceedings, Part I 34, pages 500-517. Springer, 2014. Google Scholar
  56. Hanlin Ren and Rahul Santhanam. A relativization perspective on meta-complexity. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  57. John Rompel. One-way functions are necessary and sufficient for secure signatures. In Annual ACM Symposium on Theory of Computing (STOC), pages 387-394, 1990. Google Scholar
  58. Michael Saks and Rahul Santhanam. Circuit lower bounds from NP-hardness of MCSP under Turing reductions. LIPIcs, 169, 2020. Google Scholar
  59. Michael Saks and Rahul Santhanam. On randomized reductions to the random strings. In 37th Computational Complexity Conference (CCC 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  60. 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
  61. 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.
  62. 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
  63. Luca Trevisan. Non-approximability results for optimization problems on bounded degree instances. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 453-461, 2001. Google Scholar
  64. Hoeteck Wee and Daniel Wichs. Candidate obfuscation via oblivious lwe sampling. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 127-156. Springer, 2021. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact