Single-Key to Multi-Key Functional Encryption with Polynomial Loss

  • Sanjam GargEmail author
  • Akshayaram Srinivasan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9986)


Functional encryption (FE) enables fine-grained access to encrypted data. In a FE scheme, the holder of a secret key \(\mathsf {FSK}_f\) (associated with a function f) and a ciphertext c (encrypting plaintext x) can learn f(x) but nothing more.

An important parameter in the security model for FE is the number of secret keys that adversary has access to. In this work, we give a transformation from a FE scheme for which the adversary gets access to a single secret key (with ciphertext size sub-linear in the circuit for which this secret key is issued) to one that is secure even if adversary gets access to an unbounded number of secret keys. A novel feature of our transformation is that its security proof incurs only a polynomial loss.


Random String Public Parameter Challenge Ciphertext Semantic Security Random Tape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank the anonymous TCC reviewers for useful feedback. Additionally, we thank Divya Gupta, Peihan Miao, Omkant Pandey and Mark Zhandry for insightful discussions.


  1. [ABSV15]
    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)CrossRefGoogle Scholar
  2. [AJ15]
    Ananth, P., Jain, A.: Indistinguishability obfuscation from compact functional encryption. In: Gennaro, R., Robshaw, M.J.B. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 308–326. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  3. [AJS15]
    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
  4. [AS16]
    Ananth, P., Sahai, A.: Functional encryption for turing machines. In: Kushilevitz, E., et al. (eds.) TCC 2016-A. LNCS, vol. 9562, pp. 125–153. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49096-9_6 CrossRefGoogle Scholar
  5. [BF01]
    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)CrossRefGoogle Scholar
  6. [BGI14]
    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)CrossRefGoogle Scholar
  7. [BHR12]
    Bellare, M., Hoang, V.T., Rogaway, P.: Foundations of garbled circuits. In: Yu, T., Danezis, G., Gligor, V.D. (eds.) ACM CCS 2012, October 16–18, Raleigh, NC, USA, pp. 784–796. ACM (2012)Google Scholar
  8. [BSW11]
    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)CrossRefGoogle Scholar
  9. [BV15]
    Bitansky, N., Vaikuntanathan, V.: Indistinguishability obfuscation from functional encryption. In: Guruswami, V. (ed.) 56th FOCS, October 17–20, Berkeley, CA, USA, pp. 171–190. IEEE Computer Society Press (2015)Google Scholar
  10. [BW13]
    Boneh, D., Waters, B.: Constrained pseudorandom functions and their applications. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part II. LNCS, vol. 8270, pp. 280–300. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. [Coc01]
    Cocks, C.: An Identity based encryption scheme based on quadratic residues. In: Honary, B. (ed.) Cryptography and Coding 2001. LNCS, vol. 2260, pp. 360–363. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. [GGH+13]
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th FOCS, October 26–29, Berkeley, CA, USA, pp. 40–49. IEEE Computer Society Press (2013)Google Scholar
  13. [GGM86]
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [GKP+13]
    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, June 1–4, Palo Alto, CA, USA, pp. 555–564 (2013)Google Scholar
  15. [GPS15]
    Garg, S., Pandey, O., Srinivasan, A.: On the exact cryptographic hardness of finding a nash equilibrium. Cryptology ePrint Archive, Report 2015/1078 (2015).
  16. [GPSW06]
    Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: Proceedings of the 13th ACM Conference on Computer and Communications Security, CCS 2006, October 30–November 3, Alexandria, VA, USA, pp. 89–98 (2006)Google Scholar
  17. [GPSZ16]
    Garg, S., Pandey, O., Srinivasan, A., Zhandry, M.: Breaking the sub-exponential barrier in obfustopia. Cryptology ePrint Archive, Report 2016/102 (2016).
  18. [GVW13]
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute-based encryption for circuits. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, June 1–4, Palo Alto, CA, USA, pp. 545–554. ACM Press (2013)Google Scholar
  19. [GVW15]
    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)CrossRefGoogle Scholar
  20. [HJO+15]
    Hemenway, B., Jafargholi, Z., Ostrovsky, R., Scafuro, A., Wichs, D.: Adaptively secure garbled circuits from one-way functions. IACR Cryptology ePrint Archive, 2015:1250 (2015)Google Scholar
  21. [KPTZ13]
    Kiayias, A., Papadopoulos, S., Triandopoulos, N., Zacharias, T.: Delegatable pseudorandom functions and applications. In: ACM SIGSAC Conference on Computer and Communications Security, CCS 2013, November 4–8, Berlin, Germany, pp. 669–684 (2013)Google Scholar
  22. [KSW08]
    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)CrossRefGoogle Scholar
  23. [LM16]
    Li, B., Micciancio, D.: Compactness vs collusion resistance in functional encryption. Cryptology ePrint Archive, Report 2016/561 (2016).
  24. [LP09]
    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
  25. [O’N10]
    O’Neill, A.: Definitional issues in functional encryption. IACR Cryptology ePrint Archive, 2010:556 (2010)Google Scholar
  26. [Sha84]
    Shamir, A.: Identity-based cryptosystems and signature schemes. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 47–53. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  27. [SS10]
    Sahai, A., Seyalioglu, H.: Worry-free encryption: functional encryption with public keys. In: Proceedings of the 17th ACM Conference on Computer and Communications Security, CCS 2010, October 4–8, Chicago, Illinois, USA, pp. 463–472 (2010)Google Scholar
  28. [SW05]
    Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  29. [SW14]
    Sahai, A., Waters, B.: How to use indistinguishability obfuscation: deniable encryption, and more. In: Shmoys, D.B. (ed.) 46th ACM STOC, May 31–June 3, New York, NY, USA, pp. 475–484. ACM Press (2014)Google Scholar
  30. [Wat15]
    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)CrossRefGoogle Scholar
  31. [Yao86]
    Yao, A.C.-C.: How to generate and exchange secrets (extended abstract). In: 27th FOCS, October 27–29, Toronto, Ontario, Canada, pp. 162–167. IEEE Computer Society Press (1986)Google Scholar

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA

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