Abstract
Fully homomorphic encryption (FHE) over the integers, as proposed by van Dijk et al. in 2010 and developed in a number of papers afterwards, originally supported the evaluation of Boolean circuits (i.e. mod-2 arithmetic circuits) only. It is easy to generalize the somewhat homomorphic versions of the corresponding schemes to support arithmetic operations modulo Q for any \(Q>2\), but bootstrapping those generalized variants into fully homomorphic schemes is not easy. Thus, Nuida and Kurosawa settled a significant open problem in 2015 by showing that one could in fact construct FHE over the integers with message space \({\mathbb Z}/Q{\mathbb Z}\) for any constant prime Q.
As a result of their work, we now have two different ways of homomorphically evaluating a mod-Q arithmetic circuit with an FHE scheme over the integers: one could either use their scheme with message space \({\mathbb Z}/Q{\mathbb Z}\) directly, or one could first convert the arithmetic circuit to a Boolean one, and evaluate that converted circuit using an FHE scheme with binary message space. In this paper, we compare both approaches and show that the latter is often preferable to the former.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
One can ask whether it could be beneficial to choose parameters in such a way that refreshed ciphertexts support not just one but several levels of multiplication before another bootstrapping is required. The answer is no: to support d additional levels of multiplications, one needs to increase the size \(\eta \) of the secret prime p by a factor \(2^d\), and hence overall ciphertext size \(\gamma \) by a factor \(\varOmega (2^{2d})\). This makes all operations on ciphertexts at least \(\varOmega (2^{2d})\) slower, while one gains a factor at most \(O(2^d)\) on the number of required bootstrapping operations, so there is a net efficiency loss overall.
References
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
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
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
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
Fürer, M.: Faster integer multiplication. SIAM J. Comput. 39(3), 979–1005 (2009). doi:10.1137/070711761
Gentry. C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009). crypto.stanford.edu/craig
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) STOC 2009, pp. 169–178. ACM (2009). doi:10.1145/1536414.1536440
Nuida, K., Kurosawa, K.: (Batch) fully homomorphic encryption over integers for non-binary message spaces. In: Oswald, E., Fischlin, M. (eds.) Advances in Cryptology–EUROCRYPT 2015. LNCS, vol. 9056, pp. 537–555. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46800-5_21
Pisa, P.S., Abdalla, M., Duarte, O.C.M.B.: Somewhat homomorphic encryption scheme for arithmetic operations on large integers. In: GIIS 2012, pp. 1–8. IEEE (2012). doi:10.1109/GIIS.2012.6466769
Rivest, R.L., Adleman, L.M., Dertouzos, M.L.: On data banks and privacy homomorphisms. In: DeMillo, R.A. (ed.) Foundations of Secure Computation, pp. 169–180. Academic Press (1978)
von zur Gathen, J., Seroussi, G.: Boolean circuits versus arithmetic circuits. Inf. Comput. 91(1), 142–154 (1991). doi:10.1016/0890-5401(91)90078-G
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Kim, E., Tibouchi, M. (2016). FHE Over the Integers and Modular Arithmetic Circuits. In: Foresti, S., Persiano, G. (eds) Cryptology and Network Security. CANS 2016. Lecture Notes in Computer Science(), vol 10052. Springer, Cham. https://doi.org/10.1007/978-3-319-48965-0_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-48965-0_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-48964-3
Online ISBN: 978-3-319-48965-0
eBook Packages: Computer ScienceComputer Science (R0)