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Multi-identity and Multi-key Leveled FHE from Learning with Errors

  • Michael Clear
  • Ciarán McGoldrick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9216)

Abstract

Gentry, Sahai and Waters recently presented the first (leveled) identity-based fully homomorphic (IBFHE) encryption scheme (CRYPTO 2013). Their scheme however only works in the single-identity setting; that is, homomorphic evaluation can only be performed on ciphertexts created with the same identity. In this work, we extend their results to the multi-identity setting and obtain a multi-identity IBFHE scheme that is selectively secure in the random oracle model under the hardness of Learning with Errors (LWE). We also obtain a multi-key fully-homomorphic encryption (FHE) scheme that is secure under LWE in the standard model. This is the first multi-key FHE based on a well-established assumption such as standard LWE. The multi-key FHE of López-Alt, Tromer and Vaikuntanathan (STOC 2012) relied on a non-standard assumption, referred to as the Decisional Small Polynomial Ratio assumption.

Notes

Acknowledgments

We would like to thank the anonymous reviewers of for their helpful comments. The authors would like to thank Fuqun Wang for pointing out errors in an earlier version of this paper.

References

  1. 1.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, p. 169 (2009)Google Scholar
  2. 2.
    Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010) Google Scholar
  3. 3.
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010) Google Scholar
  4. 4.
    Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-lwe and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 505–524. Springer, Heidelberg (2011) Google Scholar
  5. 5.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Ostrovsky, R. (ed.) FOCS 2011, pp. 97–106. IEEE, Olympia (2011)Google Scholar
  6. 6.
    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, Part I. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013) Google Scholar
  7. 7.
    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) Google Scholar
  8. 8.
    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) Google Scholar
  9. 9.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Proceedings of the Thirty-Seventh Annual ACM symposium on Theory of Computing, STOC 2005, pp. 84–93. ACM, New York, NY, USA (2005)Google Scholar
  10. 10.
    Clear, M., McGoldrick, C.: Bootstrappable identity-based fully homomorphic encryption. Cryptology ePrint Archive, report 2014/491 (2014). http://eprint.iacr.org/
  11. 11.
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: FOCS, pp. 40–49. IEEE Computer Society (2013)Google Scholar
  12. 12.
    Clear, M., McGoldrick, C.: Bootstrappable identity-based fully homomorphic encryption. In: Gritzalis, D., Kiayias, A., Askoxylakis, I. (eds.) CANS 2014. LNCS, vol. 8813, pp. 1–19. Springer, Heidelberg (2014) Google Scholar
  13. 13.
    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) Google Scholar
  14. 14.
    López-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Proceedings of the 44th Symposium on Theory of Computing, STOC 2012, pp. 1219–1234. ACM, New York, NY, USA (2012)Google Scholar
  15. 15.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC 2008: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 197–206. ACM, New York, (2008)Google Scholar
  16. 16.
    Agrawal, S., Boneh, D., Boyen, X.: Efficient lattice (H)IBE in the standard model. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 553–572. Springer, Heidelberg (2010) Google Scholar
  17. 17.
    Agrawal, S., Boneh, D., Boyen, X.: Lattice basis delegation in fixed dimension and shorter-ciphertext hierarchical IBE. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 98–115. Springer, Heidelberg (2010) Google Scholar
  18. 18.
    Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 523–552. Springer, Heidelberg (2010) Google Scholar
  19. 19.
    Clear, M., McGoldrick, C.: Multi-identity and multi-key leveled FHE from learning with errors. IACR Cryptology ePrint Archive 2014, p. 798 (2014). http://eprint.iacr.org/2014/798
  20. 20.
    Gilbert, H. (ed.): Advances in Cryptology - EUROCRYPT 2010. LNCS, vol. 6110. Springer, Heidelberg (2010)Google Scholar
  21. 21.
    Ajtai, M.: Generating hard instances of the short basis problem. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 1–9. Springer, Heidelberg (1999) Google Scholar
  22. 22.
    Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices. Cryptology ePrint Archive, report 2008/521 (2008). http://eprint.iacr.org/2008/521
  23. 23.
    Micciancio, D., Peikert, C.: Trapdoors for lattices: simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012) Google Scholar

Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  1. 1.School of Computer Science and StatisticsTrinity CollegeDublinRepublic of Ireland

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