On CCA-Secure Somewhat Homomorphic Encryption

  • Jake Loftus
  • Alexander May
  • Nigel P. Smart
  • Frederik Vercauteren
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7118)

Abstract

It is well known that any encryption scheme which supports any form of homomorphic operation cannot be secure against adaptive chosen ciphertext attacks. The question then arises as to what is the most stringent security definition which is achievable by homomorphic encryption schemes. Prior work has shown that various schemes which support a single homomorphic encryption scheme can be shown to be IND-CCA1, i.e. secure against lunchtime attacks. In this paper we extend this analysis to the recent fully homomorphic encryption scheme proposed by Gentry, as refined by Gentry, Halevi, Smart and Vercauteren. We show that the basic Gentry scheme is not IND-CCA1; indeed a trivial lunchtime attack allows one to recover the secret key. We then show that a minor modification to the variant of the somewhat homomorphic encryption scheme of Smart and Vercauteren will allow one to achieve IND-CCA1, indeed PA-1, in the standard model assuming a lattice based knowledge assumption. We also examine the security of the scheme against another security notion, namely security in the presence of ciphertext validity checking oracles; and show why CCA-like notions are important in applications in which multiple parties submit encrypted data to the “cloud” for secure processing.

References

  1. 1.
    Al-Riyami, S.S., Paterson, K.G.: Certificateless Public Key Cryptography. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 452–473. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Armknecht, F., Peter, A., Katzenbeisser, S.: A cleaner view on IND-CCA1 secure homomorphic encryption using SOAP. IACR e-print 2010/501 (2010), http://eprint.iacr.org/2010/501
  3. 3.
    Baek, J., Steinfeld, R., Zheng, Y.: Formal proofs for the security of signcryption. Journal of Cryptology 20(2), 203–235 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bellare, M., Palacio, A.: Towards Plaintext-Aware Public-Key Encryption Without Random Oracles. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 48–62. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Bellare, M., Rogaway, P.: Optimal Asymmetric Encryption. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 92–111. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  6. 6.
    Bendlin, R., Damgård, I., Orlandi, C., Zakarias, S.: Semi-Homomorphic Encryption and Multiparty Computation. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 169–188. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Bernhard, D., Cortier, V., Pereira, O., Smyth, B., Warinschi, B.: Adapting Helios for Provable Ballot Privacy. In: Atluri, V., Diaz, C. (eds.) ESORICS 2011. LNCS, vol. 6879, pp. 335–354. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Bleichenbacher, D.: Chosen Ciphertext Attacks Against Protocols based on the RSA Encryption Standard PKCS #1. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 1–12. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Cramer, R., Shoup, V.: A Practical Public Key Cryptosystem Provably Secure Against Adaptive Chosen Ciphertext Attack. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 13–25. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Damgård, I.B.: Towards Practical Public Key Systems Secure against Chosen Ciphertext Attacks. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 445–456. Springer, Heidelberg (1992)Google Scholar
  12. 12.
    Damgård, I., Groth, J., Salomonsen, G.: The theory and implementation of an electronic voting system. In: Secure Electronic Voting, pp. 77–99. Kluwer Academic Publishers (2002)Google Scholar
  13. 13.
    Dent, A.: A Designer’s Guide to KEMs. In: Paterson, K.G. (ed.) Cryptography and Coding 2003. LNCS, vol. 2898, pp. 133–151. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Symposium on Theory of Computing – STOC 2009, pp. 169–178. ACM (2009)Google Scholar
  16. 16.
    Gentry, C.: A fully homomorphic encryption scheme. PhD, Stanford University (2009)Google Scholar
  17. 17.
    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
  18. 18.
    Hu, Z.-Y., Sun, F.-C., Jiang, J.-C.: Ciphertext verification security of symmetric encryption schemes. Science in China Series F 52(9), 1617–1631 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Joye, M., Quisquater, J., Yung, M.: On the Power of Misbehaving Adversaries and Security Analysis of the Original EPOC. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 208–222. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Lipmaa, H.: On the CCA1-security of ElGamal and Damgård’s ElGamal. In: Lai, X., Yung, M., Lin, D. (eds.) Inscrypt 2010. LNCS, vol. 6584, pp. 18–35. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Manger, J.: A Chosen Ciphertext Attack on RSA Optimal Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1 v2.0. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 230–238. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Naor, M., Yung, M.: Public-key cryptosystems provably secure against chosen ciphertext attacks. In: Symposium on Theory of Computing – STOC 1990, pp. 427–437. ACM (1990)Google Scholar
  23. 23.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Symposium on Theory of Computing – STOC 2005, pp. 84–93. ACM (2005)Google Scholar
  24. 24.
    Rivest, R.L., Adleman, L., Dertouzos, M.L.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, pp. 169–177 (1978)Google Scholar
  25. 25.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. Journal ACM 56(6), 1–40 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Smart, N.P.: Errors Matter: Breaking RSA-Based PIN Encryption with Thirty Ciphertext Validity Queries. In: Pieprzyk, J. (ed.) CT-RSA 2010. LNCS, vol. 5985, pp. 15–25. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    Smyth, B., Cortier, V.: Attacking and fixing Helios: An analysis of ballot secrecy. In: IEEE Computer Security Foundations Symposium – CSF 2011 (to appear, 2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jake Loftus
    • 1
  • Alexander May
    • 2
  • Nigel P. Smart
    • 1
  • Frederik Vercauteren
    • 3
  1. 1.Dept. Computer ScienceUniversity of BristolBristolUnited Kingdom
  2. 2.Horst Görtz Institute for IT-Security, Faculty of MathematicsRuhr-University BochumGermany
  3. 3.COSIC - Electrical EngineeringKatholieke Universiteit LeuvenHeverleeBelgium

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