FE for Inner Products and Its Application to Decentralized ABE

  • Zhedong WangEmail author
  • Xiong Fan
  • Feng-Hao Liu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11443)


In this work, we revisit the primitive functional encryption (FE) for inner products and show its application to decentralized attribute-based encryption (ABE). Particularly, we derive an FE for inner products that satisfies a stronger notion, and show how to use such an FE to construct decentralized ABE for the class \(\{0,1\}\)-\(\mathsf {LSSS} \) against bounded collusions in the plain model. We formalize the FE notion and show how to achieve such an FE under the LWE or DDH assumption. Therefore, our resulting decentralized ABE can be constructed under the same standard assumptions, improving the prior construction by Lewko and Waters (Eurocrypt 2011). Finally, we also point out challenges to construct decentralized ABE for general functions by establishing a relation between such an ABE and witness encryption for general NP statements.



We would like to thank Qiang Tang, Mingsheng Wang for their helpful discussions and suggestions. We also thank the anonymous reviewers of PKC 2019 for their insightful advices. Zhedong Wang is supported by the National Key R&D Program of China-2017YFB0802202. Xiong Fan is supported in part by IBM under Agreement 4915013672 and NSF Award CNS-1561209. Feng-Hao Liu is supported by the NSF Award CNS-1657040. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsors.


  1. 1.
    Abdalla, M., Bourse, F., De Caro, A., Pointcheval, D.: Simple functional encryption schemes for inner products. In: Katz [38], pp. 733–751Google Scholar
  2. 2.
    Abdalla, M., Gay, R., Raykova, M., Wee, H.: Multi-input inner-product functional encryption from pairings. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part I. LNCS, vol. 10210, pp. 601–626. Springer, Cham (2017). Scholar
  3. 3.
    Agrawal, S.: Stronger security for reusable garbled circuits, general definitions and attacks. In: Katz and Shacham [40], pp. 3–35Google Scholar
  4. 4.
    Agrawal, S., Bhattacherjee, S., Phan, D.H., Stehlé, D., Yamada, S.: Efficient public trace and revoke from standard assumptions: extended abstract. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 2277–2293. ACM Press, New York (2017)Google Scholar
  5. 5.
    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). Scholar
  6. 6.
    Agrawal, S., Libert, B., Stehlé, D.: Fully secure functional encryption for inner products, from standard assumptions. In: Robshaw and Katz [50], pp. 333–362Google Scholar
  7. 7.
    Agrawal, S., Rosen, A.: Functional encryption for bounded collusions, revisited. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part I. LNCS, vol. 10677, pp. 173–205. Springer, Cham (2017). Scholar
  8. 8.
    Ananth, P., Fan, X.: Attribute based encryption for RAMs from LWE. Cryptology ePrint Archive, Report 2018/273 (2018).
  9. 9.
    Bellare, M., Hoang, V.T.: Adaptive witness encryption and asymmetric password-based cryptography. In: Katz [38], pp. 308–331Google Scholar
  10. 10.
    Benhamouda, F., Bourse, F., Lipmaa, H.: CCA-secure inner-product functional encryption from projective hash functions. In: Fehr, S. (ed.) PKC 2017, Part II. LNCS, vol. 10175, pp. 36–66. Springer, Heidelberg (2017). Scholar
  11. 11.
    Bethencourt, J., Sahai, A., Waters, B.: Ciphertext-policy attribute-based encryption. In: 2007 IEEE Symposium on Security and Privacy, pp. 321–334. IEEE Computer Society Press, May 2007Google Scholar
  12. 12.
    Bishop, A., Jain, A., Kowalczyk, L.: Function-hiding inner product encryption. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 470–491. Springer, Heidelberg (2015). Scholar
  13. 13.
    Boneh, D., et al.: Threshold cryptosystems from threshold fully homomorphic encryption. Cryptology ePrint Archive, Report 2017/956 (2017).
  14. 14.
    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). Scholar
  15. 15.
    Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.): 45th ACM STOC. ACM Press, New York (2013)Google Scholar
  16. 16.
    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). Scholar
  17. 17.
    Boyen, X.: Attribute-based functional encryption on lattices. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 122–142. Springer, Heidelberg (2013). Scholar
  18. 18.
    Boyen, X., Li, Q.: Turing machines with shortcuts: efficient attribute-based encryption for bounded functions. In: Manulis, M., Sadeghi, A.-R., Schneider, S. (eds.) ACNS 2016. LNCS, vol. 9696, pp. 267–284. Springer, Cham (2016). Scholar
  19. 19.
    Brakerski, Z., Jain, A., Komargodski, I., Passelegue, A., Wichs, D.: Non-trivial witness encryption and null-iO from standard assumptions. Cryptology ePrint Archive, Report 2017/874 (2017).
  20. 20.
    Brakerski, Z., Vaikuntanathan, V.: Circuit-ABE from LWE: unbounded attributes and semi-adaptive security. In: Robshaw and Katz [50], pp. 363–384Google Scholar
  21. 21.
    Canetti, R., Garay, J.A. (eds.): CRYPTO 2013, Part II. LNCS, vol. 8043. Springer, Heidelberg (2013). Scholar
  22. 22.
    Chase, M.: Multi-authority attribute based encryption. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 515–534. Springer, Heidelberg (2007). Scholar
  23. 23.
    Chase, M., Chow, S.S.M.: Improving privacy and security in multi-authority attribute-based encryption. In: Al-Shaer, E., Jha, S., Keromytis, A.D. (eds.) ACM CCS 2009, pp. 121–130. ACM Press, New York (2009)Google Scholar
  24. 24.
    Datta, P., Okamoto, T., Tomida, J.: Full-hiding (unbounded) multi-input inner product functional encryption from the k-linear assumption. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part II. LNCS, vol. 10770, pp. 245–277. Springer, Cham (2018). Scholar
  25. 25.
    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. IEEE Computer Society Press, October 2013Google Scholar
  26. 26.
    Garg, S., Gentry, C., Halevi, S., Sahai, A., Waters, B.: Attribute-based encryption for circuits from multilinear maps. In: Canetti and Garay [21], pp. 479–499Google Scholar
  27. 27.
    Garg, S., Gentry, C., Halevi, S., Zhandry, M.: Functional encryption without obfuscation. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016, Part II. LNCS, vol. 9563, pp. 480–511. Springer, Heidelberg (2016). Scholar
  28. 28.
    Garg, S., Gentry, C., Sahai, A., Waters, B.: Witness encryption and its applications. In: Boneh et al. [15], pp. 467–476Google Scholar
  29. 29.
    Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: How to run turing machines on encrypted data. In: Canetti and Garay [21], pp. 536–553Google Scholar
  30. 30.
    Goldwasser, S., Micali, S., Rivest, R.L.: A digital signature scheme secure against adaptive chosen-message attacks. SIAM J. Comput. 17(2), 281–308 (1988)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Functional encryption with bounded collusions via multi-party computation. In: Safavi-Naini and Canetti [52], pp. 162–179Google Scholar
  32. 32.
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute-based encryption for circuits. In: Boneh et al. [15], pp. 545–554Google Scholar
  33. 33.
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Predicate encryption for circuits from LWE. In: Gennaro, R., Robshaw, M.J.B. (eds.) CRYPTO 2015, Part II. LNCS, vol. 9216, pp. 503–523. Springer, Heidelberg (2015). Scholar
  34. 34.
    Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: Juels, A., Wright, R.N., Vimercati, S. (eds.) ACM CCS 2006, pp. 89–98. ACM Press, October/November 2006. Cryptology ePrint Archive Report 2006/309Google Scholar
  35. 35.
    Green, M., Hohenberger, S., Waters, B.: Outsourcing the decryption of ABE ciphertexts. In: USENIX Security Symposium, vol. 2011 (2011)Google Scholar
  36. 36.
    Jafargholi, Z., Kamath, C., Klein, K., Komargodski, I., Pietrzak, K., Wichs, D.: Be adaptive, avoid overcommitting. In: Katz and Shacham [40], pp. 133–163Google Scholar
  37. 37.
    Katsumata, S., Yamada, S.: Partitioning via non-linear polynomial functions: more compact IBEs from ideal lattices and bilinear maps. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016, Part II. LNCS, vol. 10032, pp. 682–712. Springer, Heidelberg (2016). Scholar
  38. 38.
    Katz, J. (ed.): PKC 2015. LNCS, vol. 9020. Springer, Heidelberg (2015). Scholar
  39. 39.
    Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 146–162. Springer, Heidelberg (2008). Scholar
  40. 40.
    Katz, J., Shacham, H. (eds.): CRYPTO 2017, Part I. LNCS, vol. 10401. Springer, Cham (2017). Scholar
  41. 41.
    Lewko, A., Waters, B.: Decentralizing attribute-based encryption. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 568–588. Springer, Heidelberg (2011). Scholar
  42. 42.
    Lewko, A.B., 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). Scholar
  43. 43.
    Lin, H., Cao, Z., Liang, X., Shao, J.: Secure threshold multi authority attribute based encryption without a central authority. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 426–436. Springer, Heidelberg (2008). Scholar
  44. 44.
    Liu, Z., Cao, Z., Huang, Q., Wong, D.S., Yuen, T.H.: Fully secure multi-authority ciphertext-policy attribute-based encryption without random oracles. In: Atluri, V., Diaz, C. (eds.) ESORICS 2011. LNCS, vol. 6879, pp. 278–297. Springer, Heidelberg (2011). Scholar
  45. 45.
    Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. In: 43rd FOCS, pp. 356–365. IEEE Computer Society Press, November 2002Google Scholar
  46. 46.
    Müller, S., Katzenbeisser, S., Eckert, C.: Distributed attribute-based encryption. In: Lee, P.J., Cheon, J.H. (eds.) ICISC 2008. LNCS, vol. 5461, pp. 20–36. Springer, Heidelberg (2009). Scholar
  47. 47.
    Muller, S., Katzenbeisser, S., Eckert, C.: On multi-authority ciphertext-policy attribute-based encryption. Bull. Korean Math. Soc. 46(4), 803–819 (2009)MathSciNetCrossRefGoogle Scholar
  48. 48.
    O’Neill, A.: Definitional issues in functional encryption. Cryptology ePrint Archive, Report 2010/556 (2010).
  49. 49.
    Ostrovsky, R., Sahai, A., Waters, B.: Attribute-based encryption with non-monotonic access structures. In: Ning, P., di Vimercati, S.D.C., Syverson, P.F. (eds.) ACM CCS 2007, pp. 195–203. ACM Press, New York (2007)Google Scholar
  50. 50.
    Robshaw, M., Katz, J. (eds.): CRYPTO 2016, Part III. LNCS, vol. 9816. Springer, Heidelberg (2016). Scholar
  51. 51.
    Rouselakis, Y., Waters, B.: Practical constructions and new proof methods for large universe attribute-based encryption. In: Sadeghi, A.-R., Gligor, V.D., Yung, M. (eds.) ACM CCS 2013, pp. 463–474. ACM Press, New York (2013)Google Scholar
  52. 52.
    Safavi-Naini, R., Canetti, R. (eds.): CRYPTO 2012. LNCS, vol. 7417. Springer, Heidelberg (2012). Scholar
  53. 53.
    Sahai, A., Waters, B.R.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005). Scholar
  54. 54.
    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). Scholar
  55. 55.
    Waters, B.: Functional encryption for regular languages. In: Safavi-Naini and Canetti [52], pp. 218–235Google Scholar
  56. 56.
    Xagawa, K.: Improved (hierarchical) inner-product encryption from lattices. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 235–252. Springer, Heidelberg (2013). Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.School of Cyber SecurityUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  3. 3.Cornell UniversityIthacaUSA
  4. 4.Florida Atlantic UniversityBoca RatonUSA

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