Skip to main content
SpringerLink
Log in

Attribute Based Encryption (and more) for Nondeterministic Finite Automata from LWE

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

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11693))

Included in the following conference series:

Abstract

Constructing Attribute Based Encryption (ABE) [56] for uniform models of computation from standard assumptions, is an important problem, about which very little is known. The only known ABE schemes in this setting that (i) avoid reliance on multilinear maps or indistinguishability obfuscation, (ii) support unbounded length inputs and (iii) permit unbounded key requests to the adversary in the security game, are by Waters from Crypto, 2012 [57] and its variants. Waters provided the first ABE for Deterministic Finite Automata (DFA) satisfying the above properties, from a parametrized or “q-type” assumption over bilinear maps. Generalizing this construction to Nondeterministic Finite Automata (NFA) was left as an explicit open problem in the same work, and has seen no progress to date. Constructions from other assumptions such as more standard pairing based assumptions, or lattice based assumptions has also proved elusive.

In this work, we construct the first symmetric key attribute based encryption scheme for nondeterministic finite automata (NFA) from the learning with errors (LWE) assumption. Our scheme supports unbounded length inputs as well as unbounded length machines. In more detail, secret keys in our construction are associated with an NFA M of unbounded length, ciphertexts are associated with a tuple \((\mathbf {x}, m)\) where \(\mathbf {x}\) is a public attribute of unbounded length and m is a secret message bit, and decryption recovers m if and only if \(M(\mathbf {x})=1\).

Further, we leverage our ABE to achieve (restricted notions of) attribute hiding analogous to the circuit setting, obtaining the first predicate encryption and bounded key functional encryption schemes for NFA from LWE. We achieve machine hiding in the single/bounded key setting to obtain the first reusable garbled NFA from standard assumptions. In terms of lower bounds, we show that secret key functional encryption even for DFAs, with security against unbounded key requests implies indistinguishability obfuscation (\(\mathsf {iO}\)) for circuits; this suggests a barrier in achieving full fledged functional encryption for NFA.

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

Access this chapter

Log in via an institution
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

Notes

  1. 1.

    A trivial workaround would be to fix the input length to some fixed upper bound and pad all data to this bound; but this solution incurs substantial overhead (besides being inelegant).

  2. 2.

    Note that an NFA can be converted to an equivalent DFA but this transformation incurs exponential blowup in machine size.

  3. 3.

    This may be generalized to bounded key, for any a-priori fixed (polynomial) bound.

  4. 4.

    The construction is later extended to be adaptively secure rather than selectively secure (e.g., [16]), but the basic structure of the construction is unchanged.

  5. 5.

    For the knowledgeable reader, bounded key variants of reusable garbled circuits exist, for instance by applying the compiler of [38], but using this in the aforementioned construction does not work due to the structure of their construction.

  6. 6.

    Recall that we are only dealing with bounded size NFAs.

References

  1. Abdalla, M., Bourse, F., Caro, A.D., Pointcheval, D.: Simple functional encryption schemes for inner products. Cryptology ePrint Archive, Report 2015/017 (2015). http://eprint.iacr.org/ To appear in PKC’15

  2. 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 

  3. Agrawal, S.: Indistinguishability obfuscation minus multilinear maps: new methods for bootstrapping and instantiation (2018)

    Google Scholar 

  4. 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 

  5. Agrawal, S., Libert, B., Stehlé, D.: Fully secure functional encryption for inner products, from standard assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 333–362. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_12

    Chapter  Google Scholar 

  6. Agrawal, S., Maitra, M.: FE and iO for turing machines from minimal assumptions. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018. LNCS, vol. 11240, pp. 473–512. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03810-6_18

    Chapter  Google Scholar 

  7. 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 

  8. Agrawal, S., Singh, I.P.: Reusable garbled deterministic finite automata from learning with errors. In: ICALP, vol. 80. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)

    Google Scholar 

  9. 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_32

    Chapter  MATH  Google Scholar 

  10. Ananth, P., Fan, X.: Attribute based encryption with sublinear decryption from LWE. Cryptology ePrint Archive, Report 2018/273 (2018). https://eprint.iacr.org/2018/273

  11. 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)

    Google Scholar 

  12. 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_15

    Chapter  Google Scholar 

  13. Ananth, P., Jain, A., Sahai, A.: Achieving compactness generically: indistinguishability obfuscation from non-compact functional encryption. IACR Cryptology ePrint Archive 2015/730 (2015)

    Google Scholar 

  14. 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 

  15. Apon, D., Döttling, N., Garg, S., Mukherjee, P.: Cryptanalysis of indistinguishability obfuscations of circuits over GGH13. eprint 2016 (2016)

    Google Scholar 

  16. Attrapadung, N.: Dual system encryption via doubly selective security: framework, fully secure functional encryption for regular languages, and more. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 557–577. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_31

    Chapter  Google Scholar 

  17. 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 

  18. 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 

  19. Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci. 38(1), 150–164 (1989)

    Article  MathSciNet  Google Scholar 

  20. Bethencourt, J., Sahai, A., Waters, B.: Ciphertext-policy attribute-based encryption. In: IEEE Symposium on Security and Privacy, pp. 321–334 (2007)

    Google Scholar 

  21. Bitansky, N., Vaikuntanathan, V.: Indistinguishability obfuscation from functional encryption. In: FOCS 2015, 163 (2015). http://eprint.iacr.org/2015/163

  22. Boneh, D., et al.: Fully key-homomorphic encryption, arithmetic circuit ABE and compact garbled circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 533–556. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_30

    Chapter  Google Scholar 

  23. Boneh, D., Waters, B.: Conjunctive, subset, and range queries on encrypted data. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 535–554. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70936-7_29

    Chapter  Google Scholar 

  24. Boyen, X., Li, Q.: Attribute-based encryption for finite automata from LWE. In: Au, M.-H., Miyaji, A. (eds.) ProvSec 2015. LNCS, vol. 9451, pp. 247–267. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26059-4_14

    Chapter  Google Scholar 

  25. Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014 (2014)

    Google Scholar 

  26. Brakerski, Z., Vaikuntanathan, V.: Circuit-ABE from LWE: unbounded attributes and semi-adaptive security. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 363–384. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_13

    Chapter  Google Scholar 

  27. 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 

  28. Cheon, J.H., Fouque, P.A., Lee, C., Minaud, B., Ryu, H.: Cryptanalysis of the new CLT multilinear map over the integers. Eprint 2016/135

    Google Scholar 

  29. Cheon, J.H., Jeong, J., Lee, C.: An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low level encoding of zero. Eprint 2016/139 (2016)

    Google Scholar 

  30. Cook, S.A., Hoover, H.J.: A depth-universal circuit. SIAM J. Comput. 14(4), 833–839 (1985). https://doi.org/10.1137/0214058

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

  32. Coron, J.S., Lee, M.S., Lepoint, T., Tibouchi, M.: Zeroizing attacks on indistinguishability obfuscation over CLT13. Eprint 2016 (2016)

    Google Scholar 

  33. 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 

  34. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: FOCS (2013). http://eprint.iacr.org/

  35. Garg, S., Gentry, C., Halevi, S., Sahai, A., Waters, B.: Attribute-based encryption for circuits from multilinear maps. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 479–499. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_27

    Chapter  Google Scholar 

  36. Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: How to run turing machines on encrypted data. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 536–553. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_30

    Chapter  Google Scholar 

  37. Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: Reusable garbled circuits and succinct functional encryption. In: STOC, pp. 555–564 (2013)

    Google Scholar 

  38. 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 

  39. Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute based encryption for circuits. In: STOC (2013)

    Google Scholar 

  40. 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 

  41. Gorbunov, S., Vinayagamurthy, D.: Riding on asymmetry: efficient abe for branching programs. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 550–574. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_23

    Chapter  Google Scholar 

  42. Goyal, R., Koppula, V., Waters, B.: Semi-adaptive security and bundling functionalities made generic and easy. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 361–388. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_14

    Chapter  Google Scholar 

  43. Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: ACM Conference on Computer and Communications Security, pp. 89–98 (2006)

    Google Scholar 

  44. Hu, Y., Jia, H.: Cryptanalysis of GGH map. Cryptology ePrint Archive: Report 2015/301 (2015)

    Google Scholar 

  45. Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 146–162. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_9

    Chapter  Google Scholar 

  46. Kitagawa, F., Nishimaki, R., Tanaka, K.: Indistinguishability obfuscation for all circuits from secret-key functional encryption. IACR Cryptology ePrint Archive 2017/361 (2017)

    Google Scholar 

  47. Kitagawa, F., Nishimaki, R., Tanaka, K.: Obfustopia built on secret-key functional encryption. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 603–648. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_20

    Chapter  Google Scholar 

  48. Kitagawa, F., Nishimaki, R., Tanaka, K., Yamakawa, T.: Adaptively secure and succinct functional encryption: Improving security and efficiency, simultaneously. Cryptology ePrint Archive, Report 2018/974 (2018). https://eprint.iacr.org/2018/974

  49. Lewko, A., Okamoto, T., Sahai, A., Takashima, K., Waters, B.: Fully secure functional encryption: attribute-based encryption and (hierarchical) inner product encryption. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 62–91. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_4

    Chapter  Google Scholar 

  50. 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 

  51. 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 

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

    Google Scholar 

  53. 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 

  54. Lin, H., Vaikuntanathan, V.: Indistinguishability obfuscation from DDH-like assumptions on constant-degree graded encodings. In: FOCS (2016)

    Google Scholar 

  55. 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 

  56. 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 

  57. Waters, B.: Functional encryption for regular languages. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 218–235. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_14

    Chapter  Google Scholar 

Download references

Acknowledgement

We thank anonymous reviewers of Crypto 2019 for their helpful comments. The third author is supported by JST CREST Grant Number JPMJCR19F6 and JSPS KAKENHI Grant Number 16K16068.

Author information

Authors and Affiliations

  1. IIT Madras, Chennai, India

    Shweta Agrawal & Monosij Maitra

  2. National Institute of Advanced Industrial Science and Technology (AIST), Tokyo, Japan

    Shota Yamada

Authors
  1. Shweta Agrawal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monosij Maitra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shota Yamada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shota Yamada .

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

Agrawal, S., Maitra, M., Yamada, S. (2019). Attribute Based Encryption (and more) for Nondeterministic Finite Automata from LWE. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26951-7_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26950-0

  • Online ISBN: 978-3-030-26951-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation