Abstract
At Eurocrypt 2010, van Dijk et al. described a fully homomorphic encryption scheme (abbreviated as DGHV) over integers. It is conceptually simple but the public key size is large. After DGHV scheme was proposed, many variants of DGHV schemes with smaller public key size were proposed. In this paper, we present a multi-ciphertexts attack on a variant of the DGHV scheme with much smaller public key (abbreviated as \(HE^{RK}\)), which was proposed by Govinda Ramaiah and Vijaya Kumari at CNC 2012. Multi-ciphertexts attack considers the security of the schemes when the attacker captures a certain amount of ciphertexts. It is a common phenomena that the attacker can easily obtain enough ciphertexts in most of practical applications of fully homomorphic encryptions (even for public-key schemes). For all the four groups of the recommended parameters of \(HE^{RK}\), we can recover the plaintexts successfully if we only capture five ciphertexts. Our attack only needs to apply LLL algorithm twice on two small dimension lattices, and the data show that the plaintexts can be recovered in seconds.
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References
Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44371-2_17
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS 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). doi:10.1007/978-3-642-22792-9_29
Cheon, J.H., Coron, J.-S., Kim, J., Lee, M.S., Lepoint, T., Tibouchi, M., Yun, A.: Batch fully homomorphic encryption over the integers. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 315–335. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38348-9_20
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.: Scale-invariant fully homomorphic encryption over the integers. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 311–328. Springer, Heidelberg (2014). doi:10.1007/978-3-642-54631-0_18
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). doi:10.1007/978-3-642-22792-9_28
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). doi:10.1007/978-3-642-29011-4_30
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). doi:10.1007/978-3-642-29011-4_27
Cad\(\acute{e}\), D., Pujol, X., Stehl\(\acute{e}\), D.: FPLLL library, version 3.0, Sep 2008. http://perso.ens-lyon.fr/damien.stehle
Cheon, J.H., Stehlé, D.: Fully homomophic encryption over the integers revisited. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 513–536. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46800-5_20
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). doi:10.1007/978-3-642-13190-5_2
Gentry, C.: A fully homomorphic encryption scheme. Ph.D. Thesis, Stanford University (2009). http://crypto.stanford.edu/craig
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st ACM Symposium on Theory of Computing, STOC 2009, pp. 169–178. ACM, New York (2009)
Gentry, C.: Toward basing fully homomorphic encryption on worst-case hardness. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 116–137. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14623-7_7
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). doi:10.1007/978-3-642-20465-4_9
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. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40041-4_5
Howgrave-Graham, N.: Approximate integer common divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001). doi:10.1007/3-540-44670-2_6
Lenstra, A.K., Lenstra Jr., H.W., Lov\(\acute{a}\)sz, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)
Nguyen, P., Stern, J.: Merkle-Hellman revisited: a cryptanalysis of the Qu-Vanstone cryptosystem based on group factorizations. In: Kaliski, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 198–212. Springer, Heidelberg (1997). doi:10.1007/BFb0052236
Nguyen, P.Q., Stehl\(\acute{e}\), D.: An LLL algorithm with quadratic complexity. SIAM J. Comput. 39(3), 874–903 (2009)
Rivest, R., Adleman, L., Dertouzos, M.: On data banks and privacy homomor- phisms. In: Foundations of Secure Computation, pp. 169–180 (1978)
Ramaiah, Y.G., Kumari, G.V.: Efficient public key generation for homomorphic encryption over the integers. In: Das, V.V., Stephen, J. (eds.) CNC 2012. LNICSSITE, vol. 108, pp. 262–268. Springer, Heidelberg (2012). doi:10.1007/978-3-642-35615-5_40
Ramaiah, Y.G., Kumari, G.V.: Towards practical homomorphic encryption with efficient public key generation, pp. 10–17. ACEEE Int. J. Netw. Secur. 03(04), 1–8 (2012)
Shoup, V.: NTL, Number Theory C++ Library. http://www.shoup.net/ntl/
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). doi:10.1007/978-3-642-13013-7_25
Acknowledgments
This paper is partially supported by: 973 Program grant 2013CB834205, NSF of China under grants No. 61502269 & 61133013 & 61272035.
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Bi, J., Liu, J., Wang, X. (2017). Cryptanalysis of a Homomorphic Encryption Scheme Over Integers. In: Chen, K., Lin, D., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2016. Lecture Notes in Computer Science(), vol 10143. Springer, Cham. https://doi.org/10.1007/978-3-319-54705-3_15
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