Fully Homomorphic Encryption over the Integers with Shorter Public Keys

  • Jean-Sébastien Coron
  • Avradip Mandal
  • David Naccache
  • Mehdi Tibouchi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6841)

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.

References

  1. 1.
    Bach, E.: How to generate factored random numbers. SIAM J. Comput. 17, 179–193 (1988)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Coron, J.S., Mandal, A., Naccache, D., Tibouchi, M.: Fully Homomorphic Encryption over the Integers with Shorter Public Keys, http://eprint.iacr.org
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009), http://crypto.stanford.edu/craig
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Grandlung, T., et al.: The GNU Multiple Precision arithmetic library, Version 4.3.2 (2010), http://gmplib.org
  9. 9.
    Lidl, R., Niederreiter, H.: Finite Fields. In: Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley, Reading (1983)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Nguyen, P.Q.: Personal CommunicationGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Stein, W.A., et al.: Sage Mathematics Software (Version 4.5.3), The Sage Development Team (2010), http://www.sagemath.org
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Pujol, X., Stehlé, D., et al.: Fplll lattice reduction library, http://perso.ens-lyon.fr/xavier.pujol/fplll/
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Stehlé, D., Steinfeld, R.: Faster Fully Homomorphic Encryption. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 377–394. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    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)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2011

Authors and Affiliations

  • Jean-Sébastien Coron
    • 1
  • Avradip Mandal
    • 1
  • David Naccache
    • 2
  • Mehdi Tibouchi
    • 1
    • 2
  1. 1.Université du LuxembourgLuxembourg
  2. 2.École normale supérieureFrance

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