Advertisement

Obfustopia Built on Secret-Key Functional Encryption

  • Fuyuki Kitagawa
  • Ryo Nishimaki
  • Keisuke Tanaka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10821)

Abstract

We show that indistinguishability obfuscation (IO) for all circuits can be constructed solely from secret-key functional encryption (SKFE). In the construction, SKFE need to be able to issue a-priori unbounded number of functional keys, that is, collusion-resistant. Our strategy is to replace public-key functional encryption (PKFE) in the construction of IO proposed by Bitansky and Vaikuntanathan (FOCS 2015) with puncturable SKFE. Bitansky and Vaikuntanathan introduced the notion of puncturable SKFE and observed that the strategy works. However, it has not been clear whether we can construct puncturable SKFE without assuming PKFE. In particular, it has not been known whether puncturable SKFE is constructed from ordinary SKFE. In this work, we show that a relaxed variant of puncturable SKFE can be constructed from collusion-resistant SKFE. Moreover, we show that the relaxed variant of puncturable SKFE is sufficient for constructing IO.

In addition, we also study the relation of collusion-resistance and succinctness for SKFE. Functional encryption is said to be weakly-succinct if the size of its encryption circuit is sub-linear in the size of functions. We show that collusion-resistant SKFE can be constructed from weakly-succinct SKFE supporting only one functional key.

By combining the above two results, we show that IO for all circuits can be constructed from weakly-succinct SKFE supporting only one functional key.

Notes

Acknowledgement

The first and third authors are supported by NTT Secure Platform Laboratories, JST CREST JPMJCR14D6, JST OPERA, JSPS KAKENHI JP16H01705, JP16J10322, JP17H01695.

References

  1. 1.
    Ananth, P., Brakerski, Z., Segev, G., Vaikuntanathan, V.: From selective to adaptive security in functional encryption. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 657–677. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48000-7_32CrossRefGoogle Scholar
  2. 2.
    Ananth, P., Jain, A.: Indistinguishability obfuscation from compact functional encryption. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 308–326. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47989-6_15CrossRefGoogle Scholar
  3. 3.
    Ananth, P., Jain, A., Sahai, A.: Indistinguishability obfuscation from functional encryption for simple functions. Cryptology ePrint Archive, Report 2015/730Google Scholar
  4. 4.
    Ananth, P., Sahai, A.: Projective arithmetic functional encryption and indistinguishability obfuscation from degree-5 multilinear maps. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 152–181. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56620-7_6CrossRefGoogle Scholar
  5. 5.
    Ananth, P.V., Gupta, D., Ishai, Y., Sahai, A.: Optimizing obfuscation: avoiding Barrington’s theorem. In: ACM CCS 2014, pp. 646–658 (2014)Google Scholar
  6. 6.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally private randomizing polynomials and their applications. Comput. Complex. 15(2), 115–162 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Apon, D., Döttling, N., Garg, S., Mukherjee, P.: Cryptanalysis of indistinguishability obfuscations of circuits over GGH13. In: 44rd International Colloquium on Automata, Languages, and Programming, ICALP 2017, Warsaw, Poland, 10–14 July 2017 (2017, to appear)Google Scholar
  8. 8.
    Applebaum, B., Brakerski, Z.: Obfuscating circuits via composite-order graded encoding. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 528–556. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46497-7_21CrossRefGoogle Scholar
  9. 9.
    Asharov, G., Segev, G.: Limits on the power of indistinguishability obfuscation and functional encryption. In: 56th FOCS, pp. 191–209 (2015)Google Scholar
  10. 10.
    Badrinarayanan, S., Miles, E., Sahai, A., Zhandry, M.: Post-zeroizing obfuscation: new mathematical tools, and the case of evasive circuits. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 764–791. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_27CrossRefGoogle Scholar
  11. 11.
    Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 221–238. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-55220-5_13CrossRefGoogle Scholar
  12. 12.
    Barak, B., Goldreich, O., Impagliazzo, R., Rudich, S., Sahai, A., Vadhan, S., Yang, K.: On the (im)possibility of obfuscating programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 1–18. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44647-8_1CrossRefGoogle Scholar
  13. 13.
    Bellare, M., Hoang, V.T., Rogaway, P.: Foundations of garbled circuits. In: ACM CCS 2012, pp. 784–796 (2012)Google Scholar
  14. 14.
    Bitansky, N., Garg, S., Lin, H., Pass, R., Telang, S.: Succinct randomized encodings and their applications. In: 47th ACM STOC, pp. 439–448 (2015)Google Scholar
  15. 15.
    Bitansky, N., Nishimaki, R., Passelègue, A., Wichs, D.: From cryptomania to obfustopia through secret-key functional encryption. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 391–418. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_15CrossRefGoogle Scholar
  16. 16.
    Bitansky, N., Paneth, O., Wichs, D.: Perfect structure on the edge of chaos. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9562, pp. 474–502. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49096-9_20CrossRefGoogle Scholar
  17. 17.
    Bitansky, N., Vaikuntanathan, V.: Indistinguishability obfuscation from functional encryption. In: 56th FOCS, pp. 171–190 (2015)Google Scholar
  18. 18.
    Boneh, D., Gupta, D., Mironov, I., Sahai, A.: Hosting services on an untrusted cloud. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 404–436. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46803-6_14Google Scholar
  19. 19.
    Boneh, D., Papakonstantinou, P.A., Rackoff, C., Vahlis, Y., Waters, B.: On the impossibility of basing identity based encryption on trapdoor permutations. In: 49th FOCS, pp. 283–292 (2008)Google Scholar
  20. 20.
    Boneh, D., Sahai, A., Waters, B.: Functional encryption: definitions and challenges. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 253–273. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19571-6_16CrossRefGoogle Scholar
  21. 21.
    Boneh, D., Waters, B.: Constrained pseudorandom functions and their applications. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8270, pp. 280–300. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-42045-0_15CrossRefGoogle Scholar
  22. 22.
    Boyle, E., Goldwasser, S., Ivan, I.: Functional signatures and pseudorandom functions. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 501–519. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-54631-0_29CrossRefGoogle Scholar
  23. 23.
    Brakerski, Z., Komargodski, I., Segev, G.: Multi-input functional encryption in the private-key setting: stronger security from weaker assumptions. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 852–880. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_30CrossRefGoogle Scholar
  24. 24.
    Brakerski, Z., Rothblum, G.N.: Virtual black-box obfuscation for all circuits via generic graded encoding. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 1–25. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-54242-8_1CrossRefGoogle Scholar
  25. 25.
    Brakerski, Z., Segev, G.: Function-private functional encryption in the private-key setting. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 306–324. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46497-7_12CrossRefGoogle Scholar
  26. 26.
    Canetti, R., Holmgren, J., Jain, A., Vaikuntanathan, V.: Succinct garbling and indistinguishability obfuscation for RAM programs. In: 47th ACM STOC, pp. 429–437 (2015)Google Scholar
  27. 27.
    Canetti, R., Lin, H., Tessaro, S., Vaikuntanathan, V.: Obfuscation of probabilistic circuits and applications. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 468–497. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46497-7_19CrossRefGoogle Scholar
  28. 28.
    Chen, Y., Gentry, C., Halevi, S.: Cryptanalyses of candidate branching program obfuscators. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 278–307. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_10CrossRefGoogle Scholar
  29. 29.
    Cohen, A., Holmgren, J., Nishimaki, R., Vaikuntanathan, V., Wichs, D.: Watermarking cryptographic capabilities. In: 48th ACM STOC, pp. 1115–1127 (2016)Google Scholar
  30. 30.
    Coron, J.-S., Gentry, C., Halevi, S., Lepoint, T., Maji, H.K., Miles, E., Raykova, M., Sahai, A., Tibouchi, M.: Zeroizing without low-level zeroes: new MMAP attacks and their limitations. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 247–266. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47989-6_12CrossRefGoogle Scholar
  31. 31.
    Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Zeroizing attacks on indistinguishability obfuscation over CLT13. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10174, pp. 41–58. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-54365-8_3CrossRefGoogle Scholar
  32. 32.
    Fernando, R., Rasmussen, P.M.R., Sahai, A.: Preventing CLT attacks on obfuscation with linear overhead. Cryptology ePrint Archive, Report 2016/1070Google Scholar
  33. 33.
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th FOCS, pp. 40–49 (2013)Google Scholar
  34. 34.
    Garg, S., Miles, E., Mukherjee, P., Sahai, A., Srinivasan, A., Zhandry, M.: Secure obfuscation in a weak multilinear map model. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 241–268. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_10CrossRefGoogle Scholar
  35. 35.
    Garg, S., Pandey, O., Srinivasan, A., Zhandry, M.: Breaking the sub-exponential barrier in obfustopia. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 156–181. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_6CrossRefGoogle Scholar
  36. 36.
    Garg, S., Srinivasan, A.: Single-key to multi-key functional encryption with polynomial loss. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 419–442. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_16CrossRefGoogle Scholar
  37. 37.
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Goldwasser, S., Gordon, S.D., Goyal, V., Jain, A., Katz, J., Liu, F.-H., Sahai, A., Shi, E., Zhou, H.-S.: Multi-input functional encryption. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 578–602. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-55220-5_32CrossRefGoogle Scholar
  39. 39.
    Hofheinz, D., Jager, T., Khurana, D., Sahai, A., Waters, B., Zhandry, M.: How to generate and use universal samplers. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 715–744. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53890-6_24CrossRefGoogle Scholar
  40. 40.
    Hohenberger, S., Sahai, A., Waters, B.: Replacing a random oracle: full domain hash from indistinguishability obfuscation. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 201–220. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-55220-5_12CrossRefGoogle Scholar
  41. 41.
    Impagliazzo, R.: A personal view of average-case complexity. In: Proceedings of the Tenth Annual Structure in Complexity Theory Conference, Minneapolis, Minnesota, USA, 19–22 June 1995, pp. 134–147 (1995)Google Scholar
  42. 42.
    Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: 21st ACM STOC, pp. 44–61 (1989)Google Scholar
  43. 43.
    Kiayias, A., Papadopoulos, S., Triandopoulos, N., Zacharias, T.: Delegatable pseudorandom functions and applications. In: ACM CCS 2013, pp. 669–684 (2013)Google Scholar
  44. 44.
    Kitagawa, F., Nishimaki, R., Tanaka, K.: From single-key to collusion-resistant secret-key functional encryption by leveraging succinctness. Cryptology ePrint Archive, Report 2017/638Google Scholar
  45. 45.
    Kitagawa, F., Nishimaki, R., Tanaka, K.: Indistinguishability obfuscation for all circuits from secret-key functional encryption. Cryptology ePrint Archive, Report 2017/361Google Scholar
  46. 46.
    Kitagawa, F., Nishimaki, R., Tanaka, K.: Simple and generic constructions of succinct functional encryption. Cryptology ePrint Archive, Report 2017/275 (to appear in PKC 2018)Google Scholar
  47. 47.
    Komargodski, I., Moran, T., Naor, M., Pass, R., Rosen, A., Yogev, E.: One-way functions and (im)perfect obfuscation. In: 55th FOCS, pp. 374–383 (2014)Google Scholar
  48. 48.
    Komargodski, I., Segev, G.: From minicrypt to obfustopia via private-key functional encryption. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 122–151. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56620-7_5CrossRefGoogle Scholar
  49. 49.
    Koppula, V., Lewko, A.B., Waters, B.: Indistinguishability obfuscation for turing machines with unbounded memory. In: 47th ACM STOC, pp. 419–428 (2015)Google Scholar
  50. 50.
    Li, B., Micciancio, D.: Compactness vs collusion resistance in functional encryption. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 443–468. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_17CrossRefGoogle Scholar
  51. 51.
    Lin, H.: Indistinguishability obfuscation from constant-degree graded encoding schemes. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 28–57. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49890-3_2CrossRefGoogle Scholar
  52. 52.
    Lin, H.: Indistinguishability obfuscation from SXDH on 5-linear maps and locality-5 PRGs. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 599–629. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_20CrossRefGoogle Scholar
  53. 53.
    Lin, H., Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation with non-trivial efficiency. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016. LNCS, vol. 9615, pp. 447–462. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49387-8_17CrossRefGoogle Scholar
  54. 54.
    Luby, M., Rackoff, C.: How to construct pseudorandom permutations from pseudorandom functions. SIAM J. Comput. 17(2), 373–386 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Lin, H., Tessaro, S.: Indistinguishability obfuscation from trilinear maps and block-wise local PRGs. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 630–660. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_21CrossRefGoogle Scholar
  56. 56.
    Lin, H., Vaikuntanathan, V.: Indistinguishability obfuscation from DDH-like assumptions on constant-degree graded encodings. In: 57th FOCS, pp. 11–20 (2016)Google Scholar
  57. 57.
    Lindell, Y., Pinkas, B.: A proof of security of yao’s protocol for two-party computation. J. Cryptol. 22(2), 161–188 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Miles, E., Sahai, A., Zhandry, M.: Annihilation attacks for multilinear maps: cryptanalysis of indistinguishability obfuscation over GGH13. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 629–658. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53008-5_22CrossRefGoogle Scholar
  59. 59.
    O’Neill, A.: Definitional issues in functional encryption. Cryptology ePrint Archive, Report 2010/556 (2010)Google Scholar
  60. 60.
    Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation from semantically-secure multilinear encodings. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 500–517. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_28CrossRefGoogle Scholar
  61. 61.
    Sahai, A., Seyalioglu, H.: Worry-free encryption: functional encryption with public keys. In: ACM CCS 2010, pp. 463–472 (2010)Google Scholar
  62. 62.
    Sahai, A., Waters, B.: How to use indistinguishability obfuscation: deniable encryption, and more. In: 46th ACM STOC, pp. 475–484 (2014)Google Scholar
  63. 63.
    Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005).  https://doi.org/10.1007/11426639_27CrossRefGoogle Scholar
  64. 64.
    Yao, A.C.-C.: How to generate and exchange secrets (extended abstract). In: 27th FOCS, pp. 162–167 (1986)Google Scholar
  65. 65.
    Zimmerman, J.: How to obfuscate programs directly. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 439–467. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46803-6_15Google Scholar

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.Secure Platform LaboratoriesNTT CorporationTokyoJapan

Personalised recommendations