Advertisement

Shift-Type Homomorphic Encryption and Its Application to Fully Homomorphic Encryption

  • Frederik Armknecht
  • Stefan Katzenbeisser
  • Andreas Peter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7374)

Abstract

This work addresses the characterization of homomorphic encryption schemes both in terms of security and design. In particular, we are interested in currently existing fully homomorphic encryption (FHE) schemes and their common structures and security. Our main contributions can be summarized as follows:

  • We define a certain type of homomorphic encryption that we call shift-type and identify it as the basic underlying structure of all existing homomorphic encryption schemes. It generalizes the already known notion of shift-type group homomorphic encryption.

  • We give an IND-CPA characterization of all shift-type homomorphic encryption schemes in terms of an abstract subset membership problem.

  • We show that this characterization carries over to all leveled FHE schemes that arise by applying Gentry’s bootstrapping technique to shift-type homomorphic encryption schemes. Since this is the common structure of all existing schemes, our result actually characterizes the IND-CPA security of all existing bootstrapping-based leveled FHE.

  • We prove that the IND-CPA security of FHE schemes that offer a certain type of circuit privacy (for FHE schemes with a binary plaintext space we require circuit privacy for a single AND-gate and, in fact, all existing binary-plaintext FHE schemes offer this) and are based on Gentry’s bootstrapping technique is equivalent to the circular security of the underlying bootstrappable scheme.

Keywords

Public-Key Cryptography Homomorphic Encryption Semantic Security Circular Security 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Melchor, C.A., Gaborit, P., Herranz, J.: Additively Homomorphic Encryption with d-Operand Multiplications. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 138–154. Springer, Heidelberg (2010)Google Scholar
  2. 2.
    Applebaum, B.: Key-Dependent Message Security: Generic Amplification and Completeness. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 527–546. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Armknecht, F., Katzenbeisser, S., Peter, A.: Group homomorphic encryption: Characterizations, impossibility results, and applications. Designs, Codes and Cryptography, 1–24, 10.1007/s10623-011-9601-2, http://dx.doi.org/10.1007/s10623-011-9601-2
  4. 4.
    Barak, B., Haitner, I., Hofheinz, D., Ishai, Y.: Bounded Key-Dependent Message Security. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 423–444. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Benaloh, J.: Verifiable secret-ballot elections. Ph.D. thesis, Yale University (1987)Google Scholar
  6. 6.
    Boneh, D., Halevi, S., Hamburg, M., Ostrovsky, R.: Circular-Secure Encryption from Decision Diffie-Hellman. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 108–125. Springer, Heidelberg (2008)Google Scholar
  7. 7.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (leveled) fully homomorphic encryption without bootstrapping. In: ITCS, pp. 309–325. ACM (2012)Google Scholar
  8. 8.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS, pp. 97–106. IEEE (2011)Google Scholar
  9. 9.
    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
  10. 10.
    Cohen, J.D., Fischer, M.J.: A robust and verifiable cryptographically secure election scheme (extended abstract). In: FOCS, pp. 372–382. IEEE (1985)Google Scholar
  11. 11.
    Cramer, R., Damgård, I., Nielsen, J.B.: Multiparty Computation from Threshold Homomorphic Encryption. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 280–299. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Cramer, R., Franklin, M.K., Schoenmakers, B., Yung, M.: Multi-authority Secret-Ballot Elections with Linear Work. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 72–83. Springer, Heidelberg (1996)Google Scholar
  13. 13.
    Cramer, R., Gennaro, R., Schoenmakers, B.: A Secure and Optimally Efficient Multi-authority Election Scheme. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 103–118. Springer, Heidelberg (1997)Google Scholar
  14. 14.
    Cramer, R., Shoup, V.: Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 45–64. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    El Gamal, T.: A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  17. 17.
    Fontaine, C., Galand, F.: A survey of homomorphic encryption for nonspecialists. EURASIP J. Inf. Secur., 15:1–15:15 (January 2007), http://dx.doi.org/10.1155/2007/13801
  18. 18.
    Gennaro, R., Gentry, C., Parno, B.: Non-interactive Verifiable Computing: Outsourcing Computation to Untrusted Workers. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 465–482. Springer, Heidelberg (2010)Google Scholar
  19. 19.
    Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009)Google Scholar
  20. 20.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178. ACM (2009)Google Scholar
  21. 21.
    Gentry, C., Halevi, S.: Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. In: FOCS, pp. 107–109. IEEE (2011)Google Scholar
  22. 22.
    Gentry, C., Halevi, S.: Implementing Gentry’s Fully-Homomorphic Encryption Scheme. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 129–148. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  23. 23.
    Gentry, C., Halevi, S., Smart, N.P.: Fully Homomorphic Encryption with Polylog Overhead. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 465–482. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Gentry, C., Halevi, S., Smart, N.P.: Better bootstrapping in fully homomorphic encryption. Cryptology ePrint Archive, Report 2011/680 (2011)Google Scholar
  25. 25.
    Ishai, Y., Paskin, A.: Evaluating Branching Programs on Encrypted Data. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 575–594. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  26. 26.
    Kushilevitz, E., Ostrovsky, R.: Replication is not needed: Single database, computationally-private information retrieval. In: FOCS, pp. 364–373 (1997)Google Scholar
  27. 27.
    Loftus, J., May, A., Smart, N.P., Vercauteren, F.: On CCA-Secure Somewhat Homomorphic Encryption. In: Miri, A., Vaudenay, S. (eds.) SAC 2011. LNCS, vol. 7118, pp. 55–72. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  28. 28.
    Naor, M., Pinkas, B.: Oblivious polynomial evaluation. SIAM J. Comput. 35(5), 1254–1281 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    Stehlé, D., Steinfeld, R.: Faster Fully Homomorphic Encryption. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 377–394. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Frederik Armknecht
    • 1
  • Stefan Katzenbeisser
    • 2
  • Andreas Peter
    • 2
  1. 1.Theoretical Computer Science and IT Security GroupUniversität MannheimGermany
  2. 2.Security Engineering GroupTechnische Universität Darmstadt and CASEDGermany

Personalised recommendations