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.
This paper is a merger of two independent works [CLT13, KLYC13] built on the same basic idea but with different contributions. The respective full versions are posted on ePrint.
Chapter PDF
Similar content being viewed by others
Keywords
References
Beneš, V.E.: Optimal rearrangeable multistage connecting networks. Bell Systems Technical Journal 43(7), 1641–1656 (1964)
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)
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS 2012, pp. 309–325. ACM (2012)
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)
Biham, E.: A fast new DES implementation in software. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 260–272. Springer, Heidelberg (1997)
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)
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)
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)
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)
Cohn, H., Heninger, N.: Approximate common divisors via lattices. Cryptology ePrint Archive, Report 2011/437 (2011), http://eprint.iacr.org
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
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)
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)
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)
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
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)
Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009), http://crypto.stanford.edu/craig
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)
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)
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)
Howgrave-Graham, N.: Approximate integer common divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)
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
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)
Lagarias, J.C.: The computational complexity of simultaneous diophantine approximation problems. SIAM J. Comput. 14(1), 196–209 (1985)
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)
Lenstra Jr., H.W.: Factoring integers with elliptic curves. The Annals of Mathematics 126(3), 649–673 (1987)
Lepoint, T., Paillier, P.: On the minimal number of bootstrappings in homomorphic circuits. In: WAHC 2013. LNCS. Springer, Heidelberg (to appear, 2013)
Memoirs of the 6th Cryptology Paper Contest, arranged by Korea Communications Commission (2012)
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)
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)
Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. To appear in Designs, Codes and Cryptography (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 International Association for Cryptologic Research
About this paper
Cite this paper
Cheon, J.H. et al. (2013). Batch Fully Homomorphic Encryption over the Integers. In: Johansson, T., Nguyen, P.Q. (eds) Advances in Cryptology – EUROCRYPT 2013. EUROCRYPT 2013. Lecture Notes in Computer Science, vol 7881. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38348-9_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-38348-9_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38347-2
Online ISBN: 978-3-642-38348-9
eBook Packages: Computer ScienceComputer Science (R0)