Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification

  • Conference paper
  • First Online:
Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11694))

Included in the following conference series:

Abstract

The existence of secure indistinguishability obfuscators (\(i\mathcal {O}\)) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing \(i\mathcal {O}\) rely on d-linear maps. While secure bilinear maps are well established in cryptographic literature, the security of candidates for \(d>2\) is poorly understood.

We propose a new approach to constructing \(i\mathcal {O}\) for general circuits. Unlike all previously known realizations of \(i\mathcal {O}\), we avoid the use of d-linear maps of degree \(d \ge 3\).

At the heart of our approach is the assumption that a new weak pseudorandom object exists. We consider two related variants of these objects, which we call perturbation resilient generator (\(\varDelta \)RG) and pseudo flawed-smudging generator (\(\mathrm {PFG}\)), respectively. At a high level, both objects are polynomially expanding functions whose outputs partially hide (or smudge) small noise vectors when added to them. We further require that they are computable by a family of degree-3 polynomials over \(\mathbb {Z}\). We show how they can be used to construct functional encryption schemes with weak security guarantees. Finally, we use novel amplification techniques to obtain full security.

As a result, we obtain \(i\mathcal {O}\) for general circuits assuming:

  • Subexponentially secure LWE

  • Bilinear Maps

  • \(\mathrm {poly}(\lambda )\)-secure 3-block-local PRGs

  • \(\varDelta \)RGs or \(\mathrm {PFG}\)s

This paper is a merge of two independent works, one by Ananth, Jain, and Sahai [AJS18], and the other by Lin and Matt [LM18].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Note that this does not necessarily mean that the resulting \(i\mathcal {O}\) constructions are insecure; in particular, there have been efforts (e.g., [GMM+16a]) in constructing \(i\mathcal {O}\) in more complex security models for multilinear maps (e.g. [MSZ16]) that have resisted polynomial-time attacks. There have also been several other \(i\mathcal {O}\) candidates proposed which are not known to polynomial-time broken (e.g. [CVW18, BGMZ18]).

  2. 2.

    A function has locality \(\ell \) if every output element depends on at most \(\ell \) input elements.

  3. 3.

    The attacks actually leave open a very small window of expansion. Nevertheless, they have weakened our confidence on the security of PRGs with block-locality 2.

  4. 4.

    n, m, p are parameterized by the security parameter \(\lambda \).

  5. 5.

    There is be a tradeoff between how much AJS can weaken the indistinguishability requirements of the \(\varDelta \)RG and the 3-block-local PRG.

  6. 6.

    In an earlier version of this paper, this overview focused on constructing degree-2 \(\varDelta \)RGs. However, as we describe now, our technical approach is more general, and we describe it in greater generality here.

  7. 7.

    The schemes in [AR17, Agr18a] has more complicated decryption equation, where the decryption noise is of form \(\sum _i p_i {{\mathbf {e}}}_i\) where \(\{p_i\}\) is a set of increasing moduli. Here we omit this complexity.

  8. 8.

    Thanks Yuval Ishai and Elette Boyal for pointing this out.

  9. 9.

    If not, one can always use garbled circuits to turn g into another circuit where every output bit is computable in size \(\mathrm {poly}(\lambda )\), at the cost of increasing the size, input length, and output length of the circuit by a multiplicative \(\mathrm {poly}(\lambda )\) factor.

References

  1. 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_21

    Chapter  Google Scholar 

  2. Agrawal, S., Freeman, D.M., Vaikuntanathan, V.: Functional encryption for inner product predicates from learning with errors. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 21–40. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_2

    Chapter  Google Scholar 

  3. Arora, S., Ge, R.: New algorithms for learning in presence of errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 403–415. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22006-7_34

    Chapter  Google Scholar 

  4. Ananth, P., Gupta, D., Ishai, Y., Sahai, A.: Optimizing obfuscation: avoiding Barrington’s theorem. In: Proceedings of the 2014 ACM SIGSAC Conference on Computer and Communications Security, CCS 2014, pp. 646–658. ACM, New York (2014)

    Google Scholar 

  5. Agrawal, S.: Stronger security for reusable garbled circuits, general definitions and attacks. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 3–35. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_1

    Chapter  Google Scholar 

  6. Agrawal, S.: New methods for indistinguishability obfuscation: bootstrapping and instantiation. Cryptology ePrint Archive, Report 2018/633 (2018). https://eprint.iacr.org/2018/633

  7. Agrawal, S.: Personal communication and a previous version of eprint report 2018/633 (2018)

    Google Scholar 

  8. Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. In: 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 166–175, October 2004

    Google Scholar 

  9. Ananth, P., Jain, A., Sahai, A.: Indistinguishability obfuscation without multilinear maps: iO from LWE, bilinear maps, and weak pseudorandomness. Cryptology ePrint Archive, Report 2018/615 (2018). https://eprint.iacr.org/2018/615

  10. Alwen, J., Krenn, S., Pietrzak, K., Wichs, D.: Learning with rounding, revisited. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 57–74. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_4

    Chapter  Google Scholar 

  11. Applebaum, B., Lovett, S.: Algebraic attacks against random local functions and their countermeasures. In: Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC 2016, pp. 1087–1100. ACM, New York (2016)

    Google Scholar 

  12. Agrawal, S., Rosen, A.: Functional encryption for bounded collusions, revisited. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 173–205. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_7

    Chapter  Google Scholar 

  13. 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_6

    Chapter  Google Scholar 

  14. Barak, B., Brakerski, Z., Komargodski, I., Kothari, P.: Limits on low-degree pseudorandom generators (or: sum-of-squares meets program obfuscation). Electron. Colloq. Comput. Complex. (ECCC) 24, 60 (2017)

    MATH  Google Scholar 

  15. Barak, B., Brakerski, Z., Komargodski, I., Kothari, P.K.: Limits on low-degree pseudorandom generators (or: sum-of-squares meets program obfuscation). In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 649–679. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_21

    Chapter  Google Scholar 

  16. Baltico, C.E.Z., Catalano, D., Fiore, D., Gay, R.: Practical functional encryption for quadratic functions with applications to predicate encryption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 67–98. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_3

    Chapter  Google Scholar 

  17. Brzuska, C., Farshim, P., Mittelbach, A.: Indistinguishability obfuscation and UCEs: the case of computationally unpredictable sources. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 188–205. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_11

    Chapter  Google Scholar 

  18. Boneh, D., et al.: Threshold cryptosystems from threshold fully homomorphic encryption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 565–596. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_19

    Chapter  Google Scholar 

  19. Brakerski, Z., Gentry, C., Halevi, S., Lepoint, T., Sahai, A., Tibouchi, M.: Cryptanalysis of the quadratic zero-testing of GGH. Cryptology ePrint Archive, Report 2015/845 (2015). http://eprint.iacr.org/

  20. Barak, B., et al.: 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_1

    Chapter  Google Scholar 

  21. Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 337–367. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_12

    Chapter  Google Scholar 

  22. 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_13

    Chapter  Google Scholar 

  23. Bartusek, J., Guan, J., Ma, F., Zhandry, M.: Return of GGH15: provable security against zeroizing attacks. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018. LNCS, vol. 11240, pp. 544–574. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03810-6_20

    Chapter  MATH  Google Scholar 

  24. Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS 2012, pp. 309–325. ACM, New York (2012)

    Google Scholar 

  25. Barak, B., Hopkins, S.B., Jain, A., Kothari, P., Sahai, A.: Sum-of-squares meets program obfuscation, revisited. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11476, pp. 226–250. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_8

    Chapter  Google Scholar 

  26. 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_27

    Chapter  Google Scholar 

  27. Bitansky, N., Paneth, O., Rosen, A.: On the cryptographic hardness of finding a nash equilibrium. In: FOCS (2015)

    Google Scholar 

  28. Bogdanov, A., Qiao, Y.: On the security of Goldreich’s one-way function. Comput. Complex. 21(1), 83–127 (2012)

    Article  MathSciNet  Google Scholar 

  29. 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_1

    Chapter  Google Scholar 

  30. Boneh, D., Silverberg, A.: Applications of multilinear forms to cryptography. Contemp. Math. 324, 71–90 (2002)

    Article  MathSciNet  Google Scholar 

  31. 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_16

    Chapter  Google Scholar 

  32. Brakerski, Z., Tsabary, R., Vaikuntanathan, V., Wee, H.: Private constrained PRFs (and more) from LWE. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 264–302. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_10

    Chapter  Google Scholar 

  33. Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 97–106, October 2011

    Google Scholar 

  34. Boneh, D., Wu, D.J., Zimmerman, J.: Immunizing multilinear maps against zeroizing attacks. IACR Cryptology ePrint Archive 2014:930 (2014)

    Google Scholar 

  35. Chen, Y.-H., Chung, K.-M., Liao, J.-J.: On the complexity of simulating auxiliary input. IACR Cryptology ePrint Archive 2018:171 (2018)

    Google Scholar 

  36. Cook, J., Etesami, O., Miller, R., Trevisan, L.: Goldreich’s one-way function candidate and myopic backtracking algorithms. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 521–538. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00457-5_31

    Chapter  Google Scholar 

  37. Coron, J.-S., et al.: 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_12

    Chapter  Google Scholar 

  38. Cheon, J.H., Han, K., Lee, C., Ryu, H., Stehlé, D.: Cryptanalysis of the multilinear map over the integers. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 3–12. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_1

    Chapter  Google Scholar 

  39. Cohen, A., Holmgren, J., Nishimaki, R., Vaikuntanathan, V., Wichs, D.: Watermarking cryptographic capabilities. In: STOC (2016)

    Google Scholar 

  40. Cheon, J.H., Lee, C., Ryu, H.: Cryptanalysis of the new CLT multilinear maps. Cryptology ePrint Archive, Report 2015/934 (2015). http://eprint.iacr.org/

  41. Coron, J.-S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_26

    Chapter  Google Scholar 

  42. Coron, J.-S., Lepoint, T., Tibouchi, M.: New multilinear maps over the integers. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 267–286. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_13

    Chapter  Google Scholar 

  43. Cryan, M., Miltersen, P.B.: On pseudorandom generators in NC\(^{0}\). In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 272–284. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44683-4_24

    Chapter  Google Scholar 

  44. Chen, Y., Vaikuntanathan, V., Wee, H.: GGH15 beyond permutation branching programs: proofs, attacks, and candidates. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 577–607. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_20

    Chapter  Google Scholar 

  45. Döttling, N., Garg, S., Gupta, D., Miao, P., Mukherjee, P.: Obfuscation from low noise multilinear maps. IACR Cryptology ePrint Archive, Report 2016/599 (2016). https://eprint.iacr.org/2016/599

  46. Goldwasser, S., et al.: 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_32

    Chapter  Google Scholar 

  47. Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_1

    Chapter  Google Scholar 

  48. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, USA, 26–29 October 2013, pp. 40–49. IEEE Computer Society (2013)

    Google Scholar 

  49. Gentry, C., Gorbunov, S., Halevi, S.: Graph-induced multilinear maps from lattices. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 498–527. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_20

    Chapter  Google Scholar 

  50. Garg, S., Gentry, C., Halevi, S., Raykova, M.: Two-round secure MPC from indistinguishability obfuscation. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 74–94. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_4

    Chapter  Google Scholar 

  51. Garg, S., Gentry, C., Halevi, S., Zhandry, M.: Functional encryption without obfuscation. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9563, pp. 480–511. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49099-0_18

    Chapter  Google Scholar 

  52. Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: Reusable garbled circuits and succinct functional encryption. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) Symposium on Theory of Computing Conference, STOC 2013, Palo Alto, CA, USA, 1–4 June 2013, pp. 555–564. ACM (2013)

    Google Scholar 

  53. Goldwasser, S., Kalai, Y.T., Peikert, C., Vaikuntanathan, V.: Robustness of the learning with errors assumption. In: Proceedings of the Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, 5–7 January 2010, pp. 230–240 (2010)

    Google Scholar 

  54. Gentry, C., Lewko, A.B., Sahai, A., Waters, B.: Indistinguishability obfuscation from the multilinear subgroup elimination assumption. IACR Cryptology ePrint Archive 2014:309 (2014)

    Google Scholar 

  55. 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_10

    Chapter  Google Scholar 

  56. Gay, R., Méaux, P., Wee, H.: Predicate encryption for multi-dimensional range queries from lattices. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 752–776. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46447-2_34

    Chapter  Google Scholar 

  57. Goldreich, O.: Candidate one-way functions based on expander graphs. Electron. Colloq. Comput. Compl. (ECCC) 7(90) (2000)

    Google Scholar 

  58. Garg, S., Pandey, O., Srinivasan, A.: Revisiting the cryptographic hardness of finding a nash equilibrium. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 579–604. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53008-5_20

    Chapter  Google Scholar 

  59. Goldwasser, S., Rothblum, G.N.: On best-possible obfuscation. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 194–213. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70936-7_11

    Chapter  Google Scholar 

  60. Grigoriev, D.: Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theor. Comput. Sci. 259(1–2), 613–622 (2001)

    Article  MathSciNet  Google Scholar 

  61. Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5

    Chapter  Google Scholar 

  62. Gorbunov, S., Vaikuntanathan, V., Wee, H.: Functional encryption with bounded collusions via multi-party computation. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 162–179. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_11

    Chapter  Google Scholar 

  63. Gorbunov, S., Vaikuntanathan, V., Wee, H.: Predicate encryption for circuits from LWE. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 503–523. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_25

    Chapter  Google Scholar 

  64. Halevi, S.: Graded encoding, variations on a scheme. IACR Cryptology ePrint Archive 2015:866 (2015)

    Google Scholar 

  65. Hu, Y., Jia, H.: Cryptanalysis of GGH map. IACR Cryptology ePrint Archive 2015:301 (2015)

    Google Scholar 

  66. 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_24

    Chapter  Google Scholar 

  67. 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_12

    Chapter  Google Scholar 

  68. Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45465-9_22

    Chapter  Google Scholar 

  69. Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: FOCS, pp. 538–545 (1995)

    Google Scholar 

  70. Jain, A., Lin, H., Matt, C., Sahai, A.: How to leverage hardness of constant-degree expanding polynomials over \(\mathbb{R}\) to build \(i\cal{O}\). In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11476, pp. 251–281. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_9

    Chapter  Google Scholar 

  71. Jain, A., Lin, H., Sahai, A.: Removing the need block-local PRGs to build iO. IACR Cryptology ePrint Archive 2019 (2019)

    Google Scholar 

  72. Jetchev, D., Pietrzak, K.: How to fake auxiliary input. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 566–590. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_24

    Chapter  Google Scholar 

  73. Jain, A., Sahai, A.: How to leverage hardness of constant-degree polynomials over R to build iO. IACR Cryptology ePrint Archive 2018:973 (2018)

    Google Scholar 

  74. Koppula, V., Lewko, A.B., Waters, B.: Indistinguishability obfuscation for turing machines with unbounded memory. In: STOC (2015)

    Google Scholar 

  75. Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48405-1_2

    Chapter  Google Scholar 

  76. 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_2

    Chapter  Google Scholar 

  77. 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_20

    Chapter  Google Scholar 

  78. Lin, H., Matt, C.: Pseudo flawed-smudging generators and their application to indistinguishability obfuscation. IACR Cryptology ePrint Archive 2018:646 (2018)

    Google Scholar 

  79. Langlois, A., Stehlé, D., Steinfeld, R.: GGHLite: more efficient multilinear maps from ideal lattices. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 239–256. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_14

    Chapter  Google Scholar 

  80. 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_21

    Chapter  Google Scholar 

  81. Lin, H., Vaikuntanathan, V.: Indistinguishability obfuscation from DDH-like assumptions on constant-degree graded encodings. In: FOCS, pp. 11–20. IEEE (2016)

    Google Scholar 

  82. Lombardi, A., Vaikuntanathan, V.: Limits on the locality of pseudorandom generators and applications to indistinguishability obfuscation. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 119–137. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_5

    Chapter  MATH  Google Scholar 

  83. Minaud, B., Fouque, P.-A.: Cryptanalysis of the new multilinear map over the integers. Cryptology ePrint Archive, Report 2015/941 (2015). http://eprint.iacr.org/

  84. Mossel, E., Shpilka, A., Trevisan, L.: On e-biased generators in NC0. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 136–145, October 2003

    Google Scholar 

  85. 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_22

    Chapter  Google Scholar 

  86. Maurer, U., Tessaro, S.: A hardcore lemma for computational indistinguishability: security amplification for arbitrarily weak PRGs with optimal stretch. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 237–254. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11799-2_15

    Chapter  Google Scholar 

  87. Mukherjee, P., Wichs, D.: Two round multiparty computation via multi-key FHE. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 735–763. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_26

    Chapter  Google Scholar 

  88. Ma, F., Zhandry, M.: The MMap strikes back: obfuscation and new multilinear maps immune to CLT13 zeroizing attacks. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018. LNCS, vol. 11240, pp. 513–543. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03810-6_19

    Chapter  Google Scholar 

  89. O’Neill, A.: Definitional issues in functional encryption. IACR Cryptology ePrint Archive 2010:556 (2010)

    Google Scholar 

  90. O’Donnell, R., Witmer, D.: Goldreich’s PRG: evidence for near-optimal polynomial stretch. In: 2014 IEEE 29th Conference on Computational Complexity (CCC), pp. 1–12, June 2014

    Google Scholar 

  91. 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_28

    Chapter  Google Scholar 

  92. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93 (2005)

    Google Scholar 

  93. Schoenebeck, G.: Linear level lasserre lower bounds for certain k-CSPs. In: 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, USA, 25–28 October 2008, pp. 593–602 (2008)

    Google Scholar 

  94. 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_27

    Chapter  Google Scholar 

  95. Sahai, A., Waters, B.: How to use indistinguishability obfuscation: deniable encryption, and more. In Shmoys, D.B. (ed) Symposium on Theory of Computing, STOC 2014, 31 May–03 June 2014, pp. 475–484. ACM, New York (2014)

    Google Scholar 

  96. Wolf, C.: “Hidden field equations” (HFE) - variations and attacks. Master’s thesis, Universität Ulm, December 2002. http://www.christopher-wolf.de/dpl

  97. 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_15

    Chapter  Google Scholar 

Download references

Acknowledgments

Huijia Lin and Christian Matt thank Shweta Agrawal for sharing an early version of [Agr18a], and Stefano Tessaro for many general discussions. This work was done in part when both of these two authors were at the University of California, Santa Barbara.

Prabhanjan Ananth, Aayush Jain and Amit Sahai thank Dakshita Khurana for collaboration in the initial stages of this research, for contributing to the writeup, and for countless discussions and comments supporting this work and improving the write up. Eventually, the current set of authors had to reluctantly agree to Dakshita’s repeated requests to not be listed in the set of authors, and hence she is in these acknowledgements instead. We thank Boaz Barak, Sam Hopkins and Pravesh Kothari for insights and extremely helpful suggestions about how attacks based on the Sum of Squares paradigm could impact our new assumptions on perturbation-resilient generators.

Huijia Lin and Christian Matt were supported by NSF grants CNS-1528178, CNS-1514526, CNS-1652849 (CAREER), a Hellman Fellowship, the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) under Contract No. W911NF-15-C-0236, and a subcontract No. 2017-002 through Galois. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government.

Prabhanjan Ananth, Aayush Jain and Amit Sahai were supported in part from a DARPA/ARL SAFEWARE award, NSF Frontier Award 1413955, and NSF grant 1619348, BSF grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. Aayush Jain is also supported by a Google PhD fellowship award in Privacy and Security. This material is based upon work supported by the Defense Advanced Research Projects Agency through the ARL under Contract W911NF-15-C- 0205. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, the U.S. Government or Google.

Author information

Authors and Affiliations

  1. MIT, Cambridge, USA

    Prabhanjan Ananth

  2. UCLA, Los Angeles, USA

    Aayush Jain & Amit Sahai

  3. University of Washington, Seattle, USA

    Huijia Lin

  4. Concordium, Zurich, Switzerland

    Christian Matt

Authors
  1. Prabhanjan Ananth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aayush Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huijia Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christian Matt
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Amit Sahai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Prabhanjan Ananth or Aayush Jain .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

Rights and permissions

Reprints and permissions

Copyright information

© 2019 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ananth, P., Jain, A., Lin, H., Matt, C., Sahai, A. (2019). Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11694. Springer, Cham. https://doi.org/10.1007/978-3-030-26954-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26954-8_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26953-1

  • Online ISBN: 978-3-030-26954-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation