On Low-End Obfuscation and Learning

Authors Elette Boyle, Yuval Ishai, Pierre Meyer, Robert Robere, Gal Yehuda

Thumbnail PDF


  • Filesize: 0.94 MB
  • 28 pages

Document Identifiers

Author Details

Elette Boyle
  • Reichman University, Herzliya, Israel
  • NTT Research, Sunnyvale, CA, USA
Yuval Ishai
  • Technion, Haifa, Israel
Pierre Meyer
  • Reichman University, Herzliya, Israel
  • IRIF, Université Paris Cité, CNRS, France
Robert Robere
  • McGill University, Montreal, Canada
Gal Yehuda
  • Technion, Haifa, Israel


We thank Geoffroy Couteau for helpful pointers and suggestions.

Cite AsGet BibTex

Elette Boyle, Yuval Ishai, Pierre Meyer, Robert Robere, and Gal Yehuda. On Low-End Obfuscation and Learning. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 23:1-23:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Most recent works on cryptographic obfuscation focus on the high-end regime of obfuscating general circuits while guaranteeing computational indistinguishability between functionally equivalent circuits. Motivated by the goals of simplicity and efficiency, we initiate a systematic study of "low-end" obfuscation, focusing on simpler representation models and information-theoretic notions of security. We obtain the following results. - Positive results via "white-box" learning. We present a general technique for obtaining perfect indistinguishability obfuscation from exact learning algorithms that are given restricted access to the representation of the input function. We demonstrate the usefulness of this approach by obtaining simple obfuscation for decision trees and multilinear read-k arithmetic formulas. - Negative results via PAC learning. A proper obfuscation scheme obfuscates programs from a class C by programs from the same class. Assuming the existence of one-way functions, we show that there is no proper indistinguishability obfuscation scheme for k-CNF formulas for any constant k ≥ 3; in fact, even obfuscating 3-CNF by k-CNF is impossible. This result applies even to computationally secure obfuscation, and makes an unexpected use of PAC learning in the context of negative results for obfuscation. - Separations. We study the relations between different information-theoretic notions of indistinguishability obfuscation, giving cryptographic evidence for separations between them.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Indistinguishability obfuscation
  • cryptography
  • learning


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


  1. Shweta Agrawal, Yuval Ishai, Eyal Kushilevitz, Varun Narayanan, Manoj Prabhakaran, Vinod M. Prabhakaran, and Alon Rosen. Secure computation from one-way noisy communication, or: Anti-correlation via anti-concentration. In Tal Malkin and Chris Peikert, editors, Advances in Cryptology - CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Virtual Event, August 16-20, 2021, Proceedings, Part II, volume 12826 of Lecture Notes in Computer Science, pages 124-154. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-84245-1_5.
  2. W. Erik Anderson. On the secure obfuscation of deterministic finite automata. Cryptology ePrint Archive, Report 2008/184, 2008. URL: https://eprint.iacr.org/2008/184.
  3. Dana Angluin. Queries and concept learning. Machine Learning, 2:319-342, 1988. URL: https://doi.org/10.1023/A:1022821128753.
  4. Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, Gaurav Rattan, and Yadu Vasudev. On the isomorphism problem for decision trees and decision lists. Theoretical Computer Science, 590:38-54, 2015. Google Scholar
  5. Boaz Barak, Nir Bitansky, Ran Canetti, Yael Tauman Kalai, Omer Paneth, and Amit Sahai. Obfuscation for evasive functions. In Yehuda Lindell, editor, TCC 2014: 11th Theory of Cryptography Conference, volume 8349 of Lecture Notes in Computer Science, pages 26-51, San Diego, CA, USA, February 24-26 2014. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-642-54242-8_2.
  6. Boaz Barak, Oded Goldreich, Russell Impagliazzo, Steven Rudich, Amit Sahai, Salil P. Vadhan, and Ke Yang. On the (im)possibility of obfuscating programs. In Joe Kilian, editor, Advances in Cryptology - CRYPTO 2001, volume 2139 of Lecture Notes in Computer Science, pages 1-18, Santa Barbara, CA, USA, August 19-23 2001. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/3-540-44647-8_1.
  7. James Bartusek, Tancrède Lepoint, Fermi Ma, and Mark Zhandry. New techniques for obfuscating conjunctions. In EUROCRYPT, pages 636-666. Springer, 2019. Google Scholar
  8. Allison Bishop, Lucas Kowalczyk, Tal Malkin, Valerio Pastro, Mariana Raykova, and Kevin Shi. A simple obfuscation scheme for pattern-matching with wildcards. In CRYPTO, pages 731-752. Springer, 2018. Google Scholar
  9. Elette Boyle, Yuval Ishai, Rafael Pass, and Mary Wootters. Can we access a database both locally and privately? In Yael Kalai and Leonid Reyzin, editors, Theory of Cryptography - 15th International Conference, TCC 2017, Baltimore, MD, USA, November 12-15, 2017, Proceedings, Part II, volume 10678 of Lecture Notes in Computer Science, pages 662-693. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-70503-3_22.
  10. Zvika Brakerski, Christina Brzuska, and Nils Fleischhacker. On statistically secure obfuscation with approximate correctness. In Annual International Cryptology Conference, pages 551-578. Springer, 2016. Google Scholar
  11. Zvika Brakerski and Guy N. Rothblum. Obfuscating conjunctions. In Ran Canetti and Juan A. Garay, editors, Advances in Cryptology - CRYPTO 2013 - 33rd Annual Cryptology Conference, Santa Barbara, CA, USA, August 18-22, 2013. Proceedings, Part II, volume 8043 of Lecture Notes in Computer Science, pages 416-434. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40084-1_24.
  12. Zvika Brakerski and Guy N. Rothblum. Black-box obfuscation for d-CNFs. In Moni Naor, editor, ITCS 2014: 5th Conference on Innovations in Theoretical Computer Science, pages 235-250, Princeton, NJ, USA, January 12-14 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2554797.2554820.
  13. Zvika Brakerski, Vinod Vaikuntanathan, Hoeteck Wee, and Daniel Wichs. Obfuscating conjunctions under entropic ring LWE. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 147-156. ACM, 2016. URL: https://doi.org/10.1145/2840728.2840764.
  14. Ran Canetti. Towards realizing random oracles: Hash functions that hide all partial information. In Burton S. Kaliski Jr., editor, Advances in Cryptology - CRYPTO '97, 17th Annual International Cryptology Conference, Santa Barbara, California, USA, August 17-21, 1997, Proceedings, volume 1294 of Lecture Notes in Computer Science, pages 455-469. Springer, 1997. URL: https://doi.org/10.1007/BFb0052255.
  15. Ran Canetti, Guy N. Rothblum, and Mayank Varia. Obfuscation of hyperplane membership. In Daniele Micciancio, editor, TCC 2010: 7th Theory of Cryptography Conference, volume 5978 of Lecture Notes in Computer Science, pages 72-89, Zurich, Switzerland, February 9-11 2010. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-642-11799-2_5.
  16. David Bruce Cousins, Giovanni Di Crescenzo, Kamil Doruk Gür, Kevin King, Yuriy Polyakov, Kurt Rohloff, Gerard W Ryan, and Erkay Savas. Implementing conjunction obfuscation under entropic ring LWE. In 2018 IEEE Symposium on Security and Privacy (SP), pages 354-371. IEEE, 2018. Google Scholar
  17. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Shai Halevi and Tal Rabin, editors, TCC 2006: 3rd Theory of Cryptography Conference, volume 3876 of Lecture Notes in Computer Science, pages 265-284, New York, NY, USA, March 4-7 2006. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/11681878_14.
  18. Andrzej Ehrenfeucht and David Haussler. Learning decision trees from random examples. Information and Computation, 82(3):231-246, 1989. Google Scholar
  19. Pooya Farshim, Georg Fuchsbauer, and Alain Passelègue. Simpler constructions of asymmetric primitives from obfuscation. In Karthikeyan Bhargavan, Elisabeth Oswald, and Manoj Prabhakaran, editors, Progress in Cryptology - INDOCRYPT 2020: 21st International Conference in Cryptology in India, volume 12578 of Lecture Notes in Computer Science, pages 715-738, Bangalore, India, December 13-16 2020. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-030-65277-7_32.
  20. Steven D. Galbraith and Lukas Zobernig. Obfuscating finite automata. Cryptology ePrint Archive, Report 2020/1009, 2020. URL: https://eprint.iacr.org/2020/1009.
  21. Sanjam Garg, Craig Gentry, Shai Halevi, Mariana Raykova, Amit Sahai, and Brent Waters. Candidate indistinguishability obfuscation and functional encryption for all circuits. SIAM J. Comput., 45(3):882-929, 2016. URL: https://doi.org/10.1137/14095772X.
  22. Craig Gentry, Shai Halevi, Mariana Raykova, and Daniel Wichs. Outsourcing private RAM computation. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 404-413. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.50.
  23. Oded Goldreich, Shafi Goldwasser, and Silvio Micali. How to construct random functions (extended abstract). In 25th Annual Symposium on Foundations of Computer Science, pages 464-479, Singer Island, Florida, October 24-26 1984. IEEE Computer Society Press. URL: https://doi.org/10.1109/SFCS.1984.715949.
  24. Oded Goldreich and Leonid A. Levin. A hard-core predicate for all one-way functions. In 21st Annual ACM Symposium on Theory of Computing, pages 25-32, Seattle, WA, USA, May 15-17 1989. ACM Press. URL: https://doi.org/10.1145/73007.73010.
  25. Shafi Goldwasser and Yael Tauman Kalai. On the impossibility of obfuscation with auxiliary input. In 46th Annual Symposium on Foundations of Computer Science, pages 553-562, Pittsburgh, PA, USA, October 23-25 2005. IEEE Computer Society Press. URL: https://doi.org/10.1109/SFCS.2005.60.
  26. Shafi Goldwasser and Silvio Micali. Probabilistic encryption. Journal of Computer and System Sciences, 28(2):270-299, 1984. Google Scholar
  27. Shafi Goldwasser and Guy N. Rothblum. On best-possible obfuscation. In Salil P. Vadhan, editor, TCC 2007: 4th Theory of Cryptography Conference, volume 4392 of Lecture Notes in Computer Science, pages 194-213, Amsterdam, The Netherlands, February 21-24 2007. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-540-70936-7_11.
  28. Rishab Goyal, Venkata Koppula, and Brent Waters. Separating semantic and circular security for symmetric-key bit encryption from the learning with errors assumption. In Jean-Sébastien Coron and Jesper Buus Nielsen, editors, Advances in Cryptology - EUROCRYPT 2017, Part II, volume 10211 of Lecture Notes in Computer Science, pages 528-557, Paris, France, April 30 - May 4 2017. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-319-56614-6_18.
  29. Satoshi Hada. Secure obfuscation for encrypted signatures. In EUROCRYPT, pages 92-112. Springer, 2010. Google Scholar
  30. Ariel Hamlin, Justin Holmgren, Mor Weiss, and Daniel Wichs. On the plausibility of fully homomorphic encryption for RAMs. In Alexandra Boldyreva and Daniele Micciancio, editors, Advances in Cryptology - CRYPTO 2019, Part I, volume 11692 of Lecture Notes in Computer Science, pages 589-619, Santa Barbara, CA, USA, August 18-22 2019. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-030-26948-7_21.
  31. Susan Hohenberger, Guy N. Rothblum, Abhi Shelat, and Vinod Vaikuntanathan. Securely obfuscating re-encryption. J. Cryptol., 24(4):694-719, 2011. URL: https://doi.org/10.1007/s00145-010-9077-7.
  32. Aayush Jain, Huijia Lin, and Amit Sahai. Indistinguishability obfuscation from well-founded assumptions. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 60-73. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451093.
  33. Michael Kearns and Leslie Valiant. Cryptographic limitations on learning boolean formulae and finite automata. J. ACM, 41(1):67-95, January 1994. URL: https://doi.org/10.1145/174644.174647.
  34. Ilan Komargodski, Tal Moran, Moni Naor, Rafael Pass, Alon Rosen, and Eylon Yogev. One-way functions and (im)perfect obfuscation. In 55th Annual Symposium on Foundations of Computer Science, pages 374-383, Philadelphia, PA, USA, October 18-21 2014. IEEE Computer Society Press. URL: https://doi.org/10.1109/FOCS.2014.47.
  35. Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, fourier transform, and learnability. J. ACM, 40(3):607-620, 1993. URL: https://doi.org/10.1145/174130.174138.
  36. Nick Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Mach. Learn., 2(4):285-318, 1987. URL: https://doi.org/10.1007/BF00116827.
  37. Michael Luby and Charles Rackoff. Pseudo-random permutation generators and cryptographic composition. In 18th Annual ACM Symposium on Theory of Computing, pages 356-363, Berkeley, CA, USA, May 28-30 1986. ACM Press. URL: https://doi.org/10.1145/12130.12167.
  38. Ben Lynn, Manoj Prabhakaran, and Amit Sahai. Positive results and techniques for obfuscation. In Christian Cachin and Jan Camenisch, editors, Advances in Cryptology - EUROCRYPT 2004, volume 3027 of Lecture Notes in Computer Science, pages 20-39, Interlaken, Switzerland, May 2-6 2004. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-540-24676-3_2.
  39. Daniele Micciancio. Oblivious data structures: Applications to cryptography. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 456-464. ACM, 1997. URL: https://doi.org/10.1145/258533.258638.
  40. Moni Naor, Gil Segev, and Udi Wieder. History-independent cuckoo hashing. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, ICALP 2008: 35th International Colloquium on Automata, Languages and Programming, Part II, volume 5126 of Lecture Notes in Computer Science, pages 631-642, Reykjavik, Iceland, July 7-11 2008. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/978-3-540-70583-3_51.
  41. Moni Naor and Vanessa Teague. Anti-presistence: History independent data structures. In 33rd Annual ACM Symposium on Theory of Computing, pages 492-501, Crete, Greece, July 6-8 2001. ACM Press. URL: https://doi.org/10.1145/380752.380844.
  42. Amit Sahai and Brent Waters. How to use indistinguishability obfuscation: Deniable encryption, and more. SIAM J. Comput., 50(3):857-908, 2021. URL: https://doi.org/10.1137/15M1030108.
  43. Victor Shoup. Lower bounds for discrete logarithms and related problems. In Walter Fumy, editor, Advances in Cryptology - EUROCRYPT'97, volume 1233 of Lecture Notes in Computer Science, pages 256-266, Konstanz, Germany, May 11-15 1997. Springer, Heidelberg, Germany. URL: https://doi.org/10.1007/3-540-69053-0_18.
  44. Amir Shpilka and Ilya Volkovich. On reconstruction and testing of read-once formulas. Theory of Computing, 10(1):465-514, 2014. Google Scholar
  45. Hans Ulrich Simon. Learning decision lists and trees with equivalence-queries. In Paul M. B. Vitányi, editor, Computational Learning Theory, Second European Conference, EuroCOLT '95, Barcelona, Spain, March 13-15, 1995, Proceedings, volume 904 of Lecture Notes in Computer Science, pages 322-336. Springer, 1995. URL: https://doi.org/10.1007/3-540-59119-2_188.
  46. L. G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, November 1984. URL: https://doi.org/10.1145/1968.1972.
  47. Hoeteck Wee. On obfuscating point functions. In Harold N. Gabow and Ronald Fagin, editors, 37th Annual ACM Symposium on Theory of Computing, pages 523-532, Baltimore, MA, USA, May 22-24 2005. ACM Press. URL: https://doi.org/10.1145/1060590.1060669.
  48. Daniel Wichs and Giorgos Zirdelis. Obfuscating compute-and-compare programs under LWE. In Chris Umans, editor, 58th Annual Symposium on Foundations of Computer Science, pages 600-611, Berkeley, CA, USA, October 15-17 2017. IEEE Computer Society Press. URL: https://doi.org/10.1109/FOCS.2017.61.