Abstract
We describe a working implementation of leveled homomorphic encryption (without bootstrapping) that can evaluate the AES-128 circuit in three different ways. One variant takes under over 36 hours to evaluate an entire AES encryption operation, using NTL (over GMP) as our underlying software platform, and running on a large-memory machine. Using SIMD techniques, we can process over 54 blocks in each evaluation, yielding an amortized rate of just under 40 minutes per block. Another implementation takes just over two and a half days to evaluate the AES operation, but can process 720 blocks in each evaluation, yielding an amortized rate of just over five minutes per block. We also detail a third implementation, which theoretically could yield even better amortized complexity, but in practice turns out to be less competitive.
For our implementations we develop both AES-specific optimizations as well as several “generic” tools for FHE evaluation. These last tools include (among others) a different variant of the Brakerski-Vaikuntanathan key-switching technique that does not require reducing the norm of the ciphertext vector, and a method of implementing the Brakerski-Gentry-Vaikuntanathan modulus-switching transformation on ciphertexts in CRT representation.
Keywords
- Evaluation Representation
- Homomorphic Encryption
- Round Function
- Noise Magnitude
- Homomorphic Evaluation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Chapter PDF
References
Boyar, J., Peralta, R.: A depth-16 circuit for the AES S-box (2011) (manuscript), http://eprint.iacr.org/2011/332
Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical GapSVP (2012) (manuscript), http://eprint.iacr.org/2012/078
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: Fully homomorphic encryption without bootstrapping. In: Innovations in Theoretical Computer Science, ITCS 2012 (2012), http://eprint.iacr.org/2011/277
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS 2011. 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)
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)
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)
Damgård, I., Keller, M.: Secure Multiparty AES. In: Sion, R. (ed.) FC 2010. LNCS, vol. 6052, pp. 367–374. Springer, Heidelberg (2010)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) STOC, pp. 169–178. ACM (2009)
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), Full version at http://eprint.iacr.org/2011/566
Huang, Y., Evans, D., Katz, J., Malka, L.: Faster secure two-party computation using garbled circuits. In: USENIX Security Symposium (2011)
Orlandi, C., Nielsen, J.B., Nordholt, P.S., Sheshank, S.: A new approach to practical active-secure two-party computation (2011) (manuscript)
Lauter, K., Naehrig, M., Vaikuntanathan, V.: Can homomorphic encryption be practical? In: CCSW, pp. 113–124. ACM (2011)
Lòpez-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: STOC. ACM (2012)
Lyubashevsky, V., Peikert, C., Regev, O.: On Ideal Lattices and Learning with Errors over Rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010)
Pinkas, B., Schneider, T., Smart, N.P., Williams, S.C.: Secure Two-Party Computation Is Practical. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 250–267. Springer, Heidelberg (2009)
Rivain, M., Prouff, E.: Provably Secure Higher-Order Masking of AES. In: Mangard, S., Standaert, F.-X. (eds.) CHES 2010. LNCS, vol. 6225, pp. 413–427. Springer, Heidelberg (2010)
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 (2011), Manuscript at http://eprint.iacr.org/2011/133
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 International Association for Cryptologic Research 2012
About this paper
Cite this paper
Gentry, C., Halevi, S., Smart, N.P. (2012). Homomorphic Evaluation of the AES Circuit. In: Safavi-Naini, R., Canetti, R. (eds) Advances in Cryptology – CRYPTO 2012. CRYPTO 2012. Lecture Notes in Computer Science, vol 7417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32009-5_49
Download citation
DOI: https://doi.org/10.1007/978-3-642-32009-5_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32008-8
Online ISBN: 978-3-642-32009-5
eBook Packages: Computer ScienceComputer Science (R0)