Abstract
At Eurocrypt 2010 van Dijk et al. described a fully homomorphic encryption scheme over the integers. The main appeal of this scheme (compared to Gentry’s) is its conceptual simplicity. This simplicity comes at the expense of a public key size in \({\cal \tilde O}(\lambda^{10})\) which is too large for any practical system. In this paper we reduce the public key size to \({\cal \tilde O}(\lambda^{7})\) by encrypting with a quadratic form in the public key elements, instead of a linear form. We prove that the scheme remains semantically secure, based on a stronger variant of the approximate-GCD problem, already considered by van Dijk et al.
We also describe the first implementation of the resulting fully homomorphic scheme. Borrowing some optimizations from the recent Gentry-Halevi implementation of Gentry’s scheme, we obtain roughly the same level of efficiency. This shows that fully homomorphic encryption can be implemented using simple arithmetic operations.
Chapter PDF
Similar content being viewed by others
Keywords
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
Bach, E.: How to generate factored random numbers. SIAM J. Comput. 17, 179–193 (1988)
Boyar, J., Peralta, R., Pochuev, D.: On the multiplicative complexity of boolean functions over the basis ( ∧ , ⊕ , 1). Theor. Comput. Sci. 235(1), 43–57 (2000)
Coron, J.S., Mandal, A., Naccache, D., Tibouchi, M.: Fully Homomorphic Encryption over the Integers with Shorter Public Keys, http://eprint.iacr.org
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. Ph.D. 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)
Goldreich, O., Goldwasser, S., Halevi, S.: Public-key cryptosystems from lattice reduction problems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112–131. Springer, Heidelberg (1997)
Grandlung, T., et al.: The GNU Multiple Precision arithmetic library, Version 4.3.2 (2010), http://gmplib.org
Lidl, R., Niederreiter, H.: Finite Fields. In: Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley, Reading (1983)
Nguyên, P.Q., Stern, J.: The Two Faces of Lattices in Cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 146–180. Springer, Heidelberg (2001)
Nguyen, P.Q.: Personal Communication
Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)
Stein, W.A., et al.: Sage Mathematics Software (Version 4.5.3), The Sage Development Team (2010), http://www.sagemath.org
Micciancio, D.: Improving Lattice Based Cryptosystems Using the Hermite Normal Form. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 126–145. Springer, Heidelberg (2001)
Pujol, X., Stehlé, D., et al.: Fplll lattice reduction library, http://perso.ens-lyon.fr/xavier.pujol/fplll/
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)
Stehlé, D., Steinfeld, R.: Faster Fully Homomorphic Encryption. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 377–394. Springer, Heidelberg (2010)
Stehlé, D., Zimmermann, P.: A binary recursive gcd algorithm. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 411–425. Springer, Heidelberg (2004)
Wegman, M.N., Carter, J.L.: New hash functions and their use in authentication and set equality. Journal of Computer and System Sciences 22(3), 265–279 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 International Association for Cryptologic Research
About this paper
Cite this paper
Coron, JS., Mandal, A., Naccache, D., Tibouchi, M. (2011). Fully Homomorphic Encryption over the Integers with Shorter Public Keys. In: Rogaway, P. (eds) Advances in Cryptology – CRYPTO 2011. CRYPTO 2011. Lecture Notes in Computer Science, vol 6841. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22792-9_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-22792-9_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22791-2
Online ISBN: 978-3-642-22792-9
eBook Packages: Computer ScienceComputer Science (R0)