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Batch Fully Homomorphic Encryption over the Integers

  • Jung Hee Cheon
  • Jean-Sébastien Coron
  • Jinsu Kim
  • Moon Sung Lee
  • Tancrède Lepoint
  • Mehdi Tibouchi
  • Aaram Yun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7881)

Abstract

We extend the fully homomorphic encryption scheme over the integers of van Dijk et al.(DGHV) into a batch fully homomorphic encryption scheme, i.e. to a scheme that supports encrypting and homomorphically processing a vector of plaintexts as a single ciphertext.

We present two variants in which the semantic security is based on different assumptions. The first variant is based on a new decisional problem, the Decisional Approximate-GCD problem, whereas the second variant is based on the more classical computational Error-Free Approximate-GCD problem but requires additional public key elements.

We also show how to perform arbitrary permutations on the underlying plaintext vector given the ciphertext and the public key. Our scheme offers competitive performance even with the bootstrapping procedure: we describe an implementation of the homomorphic evaluation of AES, with an amortized cost of about 12 minutes per AES ciphertext on a standard desktop computer; this is comparable to the timings presented by Gentry et al.at Crypto 2012 for their implementation of a Ring-LWE based fully homomorphic encryption scheme.

Keywords

Fully Homomorphic Encryption Batch Encryption Chinese Remainder Theorem Approximate GCD Homomorphic AES 

References

  1. [Ben64]
    Beneš, V.E.: Optimal rearrangeable multistage connecting networks. Bell Systems Technical Journal 43(7), 1641–1656 (1964)MATHGoogle Scholar
  2. [BGH13]
    Brakerski, Z., Gentry, C., Halevi, S.: Packed ciphertexts in LWE-based homomorphic encryption. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 1–13. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. [BGV12]
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS 2012, pp. 309–325. ACM (2012)Google Scholar
  4. [BHY09]
    Bellare, M., Hofheinz, D., Yilek, S.: Possibility and impossibility results for encryption and commitment secure under selective opening. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 1–35. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. [Bih97]
    Biham, E.: A fast new DES implementation in software. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 260–272. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  6. [BP10]
    Boyar, J., Peralta, R.: A new combinational logic minimization technique with applications to cryptology. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 178–189. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. [Bra12]
    Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical GapSVP. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012)Google Scholar
  8. [BV11a]
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 97–106. IEEE Computer Society (2011)Google Scholar
  9. [BV11b]
    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)CrossRefGoogle Scholar
  10. [CH11]
    Cohn, H., Heninger, N.: Approximate common divisors via lattices. Cryptology ePrint Archive, Report 2011/437 (2011), http://eprint.iacr.org
  11. [CLT13]
    Coron, J.-S., Lepoint, T., Tibouchi, M.: Batch fully homomorphic encryption over the integers. Cryptology ePrint Archive, Report 2013/036 (2013), http://eprint.iacr.org
  12. [CMNT11]
    Coron, J.-S., Mandal, A., Naccache, D., Tibouchi, M.: Fully homomorphic encryption over the integers with shorter public keys. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 487–504. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. [CN12]
    Chen, Y., Nguyen, P.Q.: Faster algorithms for approximate common divisors: Breaking fully-homomorphic-encryption challenges over the integers. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 502–519. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. [CNT12]
    Coron, J.-S., Naccache, D., Tibouchi, M.: Public key compression and modulus switching for fully homomorphic encryption over the integers. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 446–464. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. [CT12]
    Coron, J.-S., Tibouchi, M.: Implementation of the fully homomorphic encryption scheme over the integers with compressed public keys in sage (2012), https://github.com/coron/fhe
  16. [DGHV10]
    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
  17. [Gen09]
    Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009), http://crypto.stanford.edu/craig
  18. [GH11]
    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
  19. [GHS12a]
    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
  20. [GHS12b]
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. [HG01]
    Howgrave-Graham, N.: Approximate integer common divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. [KLYC13]
    Kim, J., Lee, M.S., Yun, A., Cheon, J.H.: CRT-based fully homomorphic encryption over the integers. Cryptology ePrint Archive, Report 2013/057 (2013), http://eprint.iacr.org
  23. [KS09]
    Käsper, E., Schwabe, P.: Faster and timing-attack resistant AES-GCM. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 1–17. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  24. [Lag85]
    Lagarias, J.C.: The computational complexity of simultaneous diophantine approximation problems. SIAM J. Comput. 14(1), 196–209 (1985)MathSciNetMATHCrossRefGoogle Scholar
  25. [LATV12]
    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 Conference, STOC 2012, pp. 1219–1234. ACM (2012)Google Scholar
  26. [Len87]
    Lenstra Jr., H.W.: Factoring integers with elliptic curves. The Annals of Mathematics 126(3), 649–673 (1987)MathSciNetMATHCrossRefGoogle Scholar
  27. [LP13]
    Lepoint, T., Paillier, P.: On the minimal number of bootstrappings in homomorphic circuits. In: WAHC 2013. LNCS. Springer, Heidelberg (to appear, 2013)Google Scholar
  28. [Mem12]
    Memoirs of the 6th Cryptology Paper Contest, arranged by Korea Communications Commission (2012)Google Scholar
  29. [NLV11]
    Naehrig, M., Lauter, K., Vaikuntanathan, V.: Can homomorphic encryption be practical? In: Proceedings of the 3rd ACM Workshop on Cloud Computing Security Workshop, CCSW 2011, pp. 113–124. ACM (2011)Google Scholar
  30. [SV10]
    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
  31. [SV11]
    Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. To appear in Designs, Codes and Cryptography (2011)Google Scholar

Copyright information

© International Association for Cryptologic Research 2013

Authors and Affiliations

  • Jung Hee Cheon
    • 1
  • Jean-Sébastien Coron
    • 2
  • Jinsu Kim
    • 1
  • Moon Sung Lee
    • 1
  • Tancrède Lepoint
    • 3
    • 4
  • Mehdi Tibouchi
    • 5
  • Aaram Yun
    • 6
  1. 1.Seoul National University (SNU)Republic of Korea
  2. 2.TranefFrance
  3. 3.CryptoExpertsFrance
  4. 4.École Normale SupérieureFrance
  5. 5.NTT Secure Platform LaboratoriesJapan
  6. 6.Ulsan National Institute of Science and Technology (UNIST)Republic of Korea

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