Advertisement

From Quadratic Functions to Polynomials: Generic Functional Encryption from Standard Assumptions

  • Linru Zhang
  • Yuechen Chen
  • Jun Zhang
  • Meiqi He
  • Siu-Ming YiuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11445)

Abstract

The “all-or-nothing” notion of traditional public-key encryptions is found to be insufficient for many emerging applications in which users are only allowed to obtain a functional value of the ciphertext without any other information about the ciphertext. Functional encryption was proposed to address this issue. However, existing functional encryption schemes for generic circuits either have bounded collusions or rely on not well studied assumptions. Recently, Abdalla et al. started a new line of work that focuses on specific functions and well-known standard assumptions. Several efficient schemes were proposed for inner-product and quadratic functions. There are still a lot of unsolved problems in this direction, in particular, whether a generic FE scheme can be constructed for quadratic functions and even higher degree polynomials. In this paper, we provide affirmative answers to these questions. First, we show an IND-secure generic functional encryption scheme against adaptive adversary for quadratic functions from standard assumptions. Second, we show how to build a functional encryption scheme for cubic functions (the first in the literature in public-key setting) from a functional encryption scheme for quadratic functions. Finally, we give a generalized method that transforms an IND-secure functional encryption scheme for degree-m polynomials to an IND-secure functional encryption scheme for degree-\((m+1)\) polynomials.

Notes

Acknowledgement

This project is partially supported by the Collaborative Research Fund (CRF) of RGC of Hong Kong (Project No. CityU C1008-16G).

Supplementary material

References

  1. 1.
    Abdalla, M., et al.: Searchable encryption revisited: consistency properties, relation to anonymous IBE, and extensions. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 205–222. Springer, Heidelberg (2005).  https://doi.org/10.1007/11535218_13CrossRefGoogle Scholar
  2. 2.
    Abdalla, M., Bourse, F., De Caro, A., Pointcheval, D.: Simple functional encryption schemes for inner products. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 733–751. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46447-2_33CrossRefGoogle Scholar
  3. 3.
    Abdalla, M., Bourse, F., De Caro, A., Pointcheval, D.: Better security for functional encryption for inner product evaluations. IACR Cryptology ePrint Archive, Report 2016/11 (2016)Google Scholar
  4. 4.
    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_12CrossRefGoogle Scholar
  5. 5.
    Apon, D., Döttling, N., Garg, S., Mukherjee, P.: Cryptanalysis of indistinguishability obfuscations of circuits over GGH13. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 80. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  6. 6.
    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_3CrossRefGoogle Scholar
  7. 7.
    Bitansky, N., Lin, H., Paneth, O.: On removing graded encodings from functional encryption. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10211, pp. 3–29. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56614-6_1CrossRefGoogle Scholar
  8. 8.
    Boneh, D., Boyen, X.: Efficient selective-ID secure identity-based encryption without random oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 223–238. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_14CrossRefGoogle Scholar
  9. 9.
    Boneh, D., Franklin, M.: Identity-based encryption from the weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44647-8_13CrossRefGoogle Scholar
  10. 10.
    Boneh, D., Raghunathan, A., Segev, G.: Function-private identity-based encryption: hiding the function in functional encryption. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 461–478. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40084-1_26CrossRefzbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    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_29CrossRefGoogle Scholar
  13. 13.
    Boyle, E., Chung, K.-M., Pass, R.: On extractability obfuscation. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 52–73. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-54242-8_3CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    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_1CrossRefGoogle Scholar
  16. 16.
    Cheon, J.H., Hong, S., Lee, C., Son, Y.: Polynomial functional encryption scheme with linear ciphertext size. IACR Cryptology ePrint Archive, Report 2018/585 (2018)Google Scholar
  17. 17.
    Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Cryptanalysis of GGH15 multilinear maps. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 607–628. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53008-5_21CrossRefGoogle Scholar
  18. 18.
    Datta, P., Dutta, R., Mukhopadhyay, S.: Functional encryption for inner product with full function privacy. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016. LNCS, vol. 9614, pp. 164–195. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49384-7_7CrossRefGoogle Scholar
  19. 19.
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. SIAM J. Comput. 45(3), 882–929 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Garg, S., Gentry, C., Halevi, S., Zhandry, M.: Fully secure attribute based encryption from multilinear maps. Cryptology ePrint Archive, Report 2014/622 (2014)Google Scholar
  21. 21.
    Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: Reusable garbled circuits and succinct functional encryption. In: Symposium on Theory of Computing Conference, STOC 2013, Palo Alto, California, USA, 1–4 June 2013, pp. 555–564. ACM (2013)Google Scholar
  22. 22.
    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_11CrossRefGoogle Scholar
  23. 23.
    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_9CrossRefGoogle Scholar
  24. 24.
    Kim, S., Lewi, K., Mandal, A., Montgomery, H.W., Roy, A., Wu, D.J.: Function-hiding inner product encryption is practical. Cryptology ePrint Archive, Report 2016/440Google Scholar
  25. 25.
    Lin, H., Vaikuntanathan, V.: Indistinguishability obfuscation from DDH-like assumptions on constant-degree graded encodings. In: Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, Brunswick, New Jersey, USA, 9–11 October 2016, pp. 11–20. IEEE (2016)Google Scholar
  26. 26.
    O’Neill, A.: Definitional issues in functional encryption. Cryptology ePrint Archive, Report 2010/556 (2010)Google Scholar
  27. 27.
    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
  28. 28.
    Shamir, A.: Identity-based cryptosystems and signature schemes. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 47–53. Springer, Heidelberg (1985).  https://doi.org/10.1007/3-540-39568-7_5CrossRefGoogle Scholar
  29. 29.
    Shen, E., Shi, E., Waters, B.: Predicate privacy in encryption systems. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 457–473. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-00457-5_27CrossRefGoogle Scholar
  30. 30.
    Waters, B.: Ciphertext-policy attribute-based encryption: an expressive, efficient, and provably secure realization. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 53–70. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19379-8_4CrossRefGoogle Scholar
  31. 31.
    Waters, B.: A punctured programming approach to adaptively secure functional encryption. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 678–697. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48000-7_33CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Linru Zhang
    • 1
  • Yuechen Chen
    • 1
  • Jun Zhang
    • 2
  • Meiqi He
    • 1
  • Siu-Ming Yiu
    • 1
    Email author
  1. 1.Department of Computer ScienceThe University of Hong KongPokfulamHong Kong SAR, China
  2. 2.Educational Technology DepartmentShenzhen UniversityShenzhenChina

Personalised recommendations