Abstract
We present a new tensoring technique for LWE-based fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically (\(B \rightarrow B^2\cdot \text {poly}(n)\)) with every multiplication (before “refreshing”), our noise only grows linearly (\(B \rightarrow B\cdot \text {poly}(n)\)).
We use this technique to construct a scale-invariant fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus q and the initial noise level B, and not on their absolute values.
Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process (no need for “modulus switching”), and this modulus can take arbitrary form. In addition, security can be classically reduced from the worst-case hardness of the GapSVP problem (with quasi-polynomial approximation factor), whereas previous constructions could only exhibit a quantum reduction from GapSVP.
Chapter PDF
Similar content being viewed by others
Keywords
- Homomorphic Encryption
- Arithmetic Circuit
- Homomorphic Encryption Scheme
- Noise Magnitude
- Invariant Perspective
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.
References
Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: Leighton, F.T., Shor, P.W. (eds.) STOC, pp. 284–293. ACM (1997)
Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast Cryptographic Primitives and Circular-Secure Encryption Based on Hard Learning Problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009)
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (leveled) fully homomorphic encryption without bootstrapping. In: ITCS (2012), http://eprint.iacr.org/2011/277
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)
Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Ostrovsky (ed.) [19], pp. 97–106, References are to full version, http://eprint.iacr.org/2011/344
Coron, J.-S., Mandal, A., Naccache, D., Tibouchi, M.: Fully homomorphic encryption over the integers with shorter public keys. In: Rogaway (ed.) [24], pp. 487–504
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.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178 (2009)
Gentry, C., Halevi, S.: Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. In: Ostrovsky (ed.) [19], pp. 107–109
Gentry, C., Halevi, S., Smart, N.P.: Better bootstrapping in fully homomorphic encryption. IACR Cryptology ePrint Archive, 2011:680 (2011)
Gentry, C., Halevi, S., Smart, N.P.: Fully homomorphic encryption with polylog overhead. IACR Cryptology ePrint Archive, 2011:566 (2011)
Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Dwork, C. (ed.) STOC, pp. 197–206. ACM (2008)
Goldreich, O., Goldwasser, S., Halevi, S.: Eliminating Decryption Errors in the Ajtai-Dwork Cryptosystem. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 105–111. Springer, Heidelberg (1997)
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); Draft of full version was provided by the authors
Micciancio, D., Mol, P.: Pseudorandom knapsacks and the sample complexity of lwe search-to-decision reductions. In: Rogaway (ed.) [24], pp. 465–484
Micciancio, D., Peikert, C.: Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012)
Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. In: Schulman, L.J. (ed.) STOC, pp. 351–358. ACM (2010)
Ostrovsky, R.: IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25. IEEE (2011)
Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In: STOC, pp. 333–342 (2009)
Regev, O.: New lattice based cryptographic constructions. In: Larmore, L.L., Goemans, M.X. (eds.) STOC, pp. 407–416. ACM (2003); Full version in J. ACM 51(6) (2004)
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 84–93. ACM (2005); Full version in J. ACM 56(6) (2009)
Rivest, R., Adleman, L., Dertouzos, M.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, pp. 169–177. Academic Press (1978)
Rogaway, P. (ed.): CRYPTO 2011. LNCS, vol. 6841. Springer, Heidelberg (2011)
Schnorr, C.-P.: A hierarchy of polynomial time lattice basis reduction algorithms. Theor. Comput. Sci. 53, 201–224 (1987)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 International Association for Cryptologic Research 2012
About this paper
Cite this paper
Brakerski, Z. (2012). Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP. 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_50
Download citation
DOI: https://doi.org/10.1007/978-3-642-32009-5_50
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)