Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes

  • N. P. Smart
  • F. Vercauteren
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6056)


We present a fully homomorphic encryption scheme which has both relatively small key and ciphertext size. Our construction follows that of Gentry by producing a fully homomorphic scheme from a “somewhat” homomorphic scheme. For the somewhat homomorphic scheme the public and private keys consist of two large integers (one of which is shared by both the public and private key) and the ciphertext consists of one large integer. As such, our scheme has smaller message expansion and key size than Gentry’s original scheme. In addition, our proposal allows efficient fully homomorphic encryption over any field of characteristic two.


Prime Ideal Element Representation Symmetric Polynomial Homomorphic Encryption Challenge Ciphertext 
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  1. 1.
    Buchmann, J.: Zur Komplexität der Berechungung von Einheiten und Klassenzahlen algebraischer Zahlkörper, Habilitationsschrift (1987)Google Scholar
  2. 2.
    Buchmann, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. Séminaire de Théorie des Nombres – Paris 1988-89, 27–41 (1990)Google Scholar
  3. 3.
    Buchmann, J., Maurer, M., Möller, B.: Cryptography based on number fields with large regulator. Journal de Théorie des Nombres de Bordeaux 12, 293–307 (2000)zbMATHGoogle Scholar
  4. 4.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Springer GTM 138 (1993)Google Scholar
  5. 5.
    Ding, J., Lindner, R.: Identifying ideal lattices. IACR eprint 2009/322Google Scholar
  6. 6.
    Von Zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  7. 7.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Symposium on Theory of Computing – STOC 2009, pp. 169–178. ACM, New York (2009)CrossRefGoogle Scholar
  8. 8.
    Gentry, C.: A fully homomorphic encryption scheme, (manuscript) (2009)Google Scholar
  9. 9.
    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
  10. 10.
    Hallgren, S.: Fast quantum algorithms for computing the unit group and class group of a number field. In: Symposium on Theory of Computing – STOC 2005, pp. 468–474. ACM, New York (2005)CrossRefGoogle Scholar
  11. 11.
    Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Ann. 261, 513–534 (1982)Google Scholar
  13. 13.
    Nguyen, 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
  14. 14.
    Thiel, C.: On the complexity of some problems in algorithmic algebraic number theory. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany (1995)Google Scholar
  15. 15.
    de Weger, B.M.M.: Algorithms for Diophantine Equations. PhD thesis, University of Leiden (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • N. P. Smart
    • 1
  • F. Vercauteren
    • 2
  1. 1.Dept. Computer ScienceUniversity of BristolBristolUnited Kingdom
  2. 2.COSIC - Electrical EngineeringKatholieke Universiteit LeuvenHeverleeBelgium

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