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On White-Box Learning and Public-Key Encryption

Authors Yanyi Liu, Noam Mazor, Rafael Pass

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Author Details

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

Funding

  • Liu, Yanyi: Research partly supported by NSF CNS-2149305.
  • Mazor, Noam: Research partly supported by NSF CNS-2149305, AFOSR Award FA9550-23-1-0312 and AFOSR Award FA9550-23-1-0387 and ISF Award 2338/23.
  • Pass, Rafael: Supported in part by NSF Award CNS 2149305, AFOSR Award FA9550-23-1-0387, AFOSR Award FA9550-23-1-0312 and AFOSR Award FA9550-24-1-0267 and ISF Award 2338/23. 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, or the AFOSR.

Cite As Get BibTex

Yanyi Liu, Noam Mazor, and Rafael Pass. On White-Box Learning and Public-Key Encryption. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 73:1-73:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.73

Abstract

We consider a generalization of the Learning With Error problem, referred to as the white-box learning problem: You are given the code of a sampler that with high probability produces samples of the form y,f(y) + ε where ε is small, and f is computable in polynomial-size, and the computational task consist of outputting a polynomial-size circuit C that with probability, say, 1/3 over a new sample y' according to the same distributions, approximates f(y') (i.e., |C(y')-f(y')| is small). This problem can be thought of as a generalizing of the Learning with Error Problem (LWE) from linear functions f to polynomial-size computable functions.
We demonstrate that worst-case hardness of the white-box learning problem, conditioned on the instances satisfying a notion of computational shallowness (a concept from the study of Kolmogorov complexity) not only suffices to get public-key encryption, but is also necessary; as such, this yields the first problem whose worst-case hardness characterizes the existence of public-key encryption. Additionally, our results highlights to what extent LWE "overshoots" the task of public-key encryption.
We complement these results by noting that worst-case hardness of the same problem, but restricting the learner to only get black-box access to the sampler, characterizes one-way functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Public-Key Encryption
  • White-Box Learning

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References

  1. Miklós Ajtai and Cynthia Dwork. A public-key cryptosystem with worst-case/average-case equivalence. In stoc29, pages 284-293, 1997. See also ECCC TR96-065. URL: https://doi.org/10.1145/258533.258604.
  2. Michael Alekhnovich. More on average case vs approximation complexity. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 298-307. IEEE, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238204.
  3. Luis Antunes and Lance Fortnow. Worst-case running times for average-case algorithms. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 298-303. IEEE, 2009. URL: https://doi.org/10.1109/CCC.2009.12.
  4. Luis Antunes, Lance Fortnow, Dieter Van Melkebeek, and N Variyam Vinodchandran. Computational depth: concept and applications. Theoretical Computer Science, 354(3):391-404, 2006. URL: https://doi.org/10.1016/J.TCS.2005.11.033.
  5. Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 171-180, 2010. URL: https://doi.org/10.1145/1806689.1806715.
  6. Marshall Ball, Yanyi Liu, Noam Mazor, and Rafael Pass. Kolmogorov comes to cryptomania: On interactive kolmogorov complexity and key-agreement. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 458-483. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00034.
  7. Avrim Blum, Merrick Furst, Michael Kearns, and Richard J Lipton. Cryptographic primitives based on hard learning problems. In Annual International Cryptology Conference, pages 278-291. Springer, 1993. URL: https://doi.org/10.1007/3-540-48329-2_24.
  8. Andrej Bogdanov, Miguel Cueto Noval, Charlotte Hoffmann, and Alon Rosen. Public-key encryption from homogeneous clwe. In Theory of Cryptography: 20th International Conference, TCC 2022, Chicago, IL, USA, November 7-10, 2022, Proceedings, Part II, pages 565-592. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22365-5_20.
  9. Zvika Brakerski, Adeline Langlois, Chris Peikert, Oded Regev, and Damien Stehlé. Classical hardness of learning with errors. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 575-584, 2013. URL: https://doi.org/10.1145/2488608.2488680.
  10. Zvika Brakerski and Vinod Vaikuntanathan. Efficient fully homomorphic encryption from (standard) lwe. Journal of the ACM, 43(2):831-871, 2014. Google Scholar
  11. Gregory J. Chaitin. On the simplicity and speed of programs for computing infinite sets of natural numbers. J. ACM, 16(3):407-422, 1969. URL: https://doi.org/10.1145/321526.321530.
  12. Whitfield Diffie and Martin E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, pages 644-654, 1976. URL: https://doi.org/10.1109/TIT.1976.1055638.
  13. Taher ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. In Annual International Cryptology Conference (CRYPTO), pages 10-18, 1984. Google Scholar
  14. Craig Gentry. Fully homomorphic encryption using ideal lattices. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 169-178, 2009. URL: https://doi.org/10.1145/1536414.1536440.
  15. Oded Goldreich and Leonid A. Levin. A hard-core predicate for all one-way functions. In Proceedings of the twenty-first annual ACM symposium on Theory of computing (STOC), pages 25-32, 1989. URL: https://doi.org/10.1145/73007.73010.
  16. Danny Harnik, Joe Kilian, Moni Naor, Omer Reingold, and Alon Rosen. On robust combiners for oblivious transfer and other primitives. In Advances in Cryptology-EUROCRYPT 2005: 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Aarhus, Denmark, May 22-26, 2005. Proceedings 24, pages 96-113. Springer, 2005. URL: https://doi.org/10.1007/11426639_6.
  17. 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.
  18. Shuichi Hirahara and Mikito Nanashima. Learning in pessiland via inductive inference. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 447-457. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00033.
  19. Thomas Holenstein. Strengthening key agreement using hard-core sets. PhD thesis, ETH Zurich, 2006. URL: https://doi.org/10.3929/ETHZ-A-005205852.
  20. Russell Impagliazzo and Leonid A. Levin. No better ways to generate hard NP instances than picking uniformly at random. In focs31, pages 812-821, 1990. URL: https://doi.org/10.1109/FSCS.1990.89604.
  21. Michael Kearns and Leslie Valiant. Cryptographic limitations on learning boolean formulae and finite automata. Journal of the ACM (JACM), 41(1):67-95, 1994. URL: https://doi.org/10.1145/174644.174647.
  22. 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.
  23. A. N. Kolmogorov. Three approaches to the quantitative definition of information. International Journal of Computer Mathematics, 2(1-4):157-168, 1968. Google Scholar
  24. Yanyi Liu, Noam Mazor, and Rafael Pass. On white-box learning and public-key encryption. Cryptology ePrint Archive, Paper 2024/1931, 2024. URL: https://eprint.iacr.org/2024/1931.
  25. 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. URL: https://doi.org/10.1109/FOCS46700.2020.00118.
  26. 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
  27. Robert J McEliece. A public-key cryptosystem based on algebraic. Coding Thv, 4244:114-116, 1978. Google Scholar
  28. Chris Peikert. Public-key cryptosystems from the worst-case shortest vector problem. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 333-342, 2009. Google Scholar
  29. Michael O Rabin. Digitalized signatures and public-key functions as intractable as factorization. Technical report, Massachusetts Inst of Tech Cambridge Lab for Computer Science, 1979. Google Scholar
  30. Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. Journal of the ACM (JACM), 56(6):1-40, 2009. URL: https://doi.org/10.1145/1568318.1568324.
  31. Ronald L Rivest, Adi Shamir, and Leonard Adleman. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2):120-126, 1978. URL: https://doi.org/10.1145/359340.359342.
  32. Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review, 41(2):303-332, 1999. URL: https://doi.org/10.1137/S0036144598347011.
  33. Michael Sipser. A complexity theoretic approach to randomness. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 330-335, 1983. URL: https://doi.org/10.1145/800061.808762.
  34. 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.
  35. Leslie G Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134-1142, 1984. URL: https://doi.org/10.1145/1968.1972.
  36. Andrew C. Yao. Theory and applications of trapdoor functions. In Annual Symposium on Foundations of Computer Science (FOCS), pages 80-91, 1982. URL: https://doi.org/10.1109/SFCS.1982.45.
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