Improved Key Generation for Gentry’s Fully Homomorphic Encryption Scheme

  • Peter Scholl
  • Nigel P. Smart
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7089)


A key problem with the original implementation of the Gentry Fully Homomorphic Encryption scheme was the slow key generation process. Gentry and Halevi provided a fast technique for 2-power cyclotomic fields. We present an extension of the Gentry–Halevi key generation technique for arbitrary cyclotomic fields. Our new method is roughly twice as efficient as the previous best methods. Our estimates are backed up with experimental data.


Fast Fourier Transform Discrete Fourier Transform Homomorphic Encryption Fast Fourier Transform Algorithm Defense Advance Research Project Agency 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cohen, H.: A Course in Computational Algebraic Number Theory, vol. 138. Springer GTM (1993)Google Scholar
  2. 2.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Symposium on Theory of Computing – STOC 2009, pp. 169–178. ACM (2009)Google Scholar
  5. 5.
    Gentry, C.: A fully homomorphic encryption scheme. PhD, Stanford University (2009)Google Scholar
  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.
    Good, I.J.: The interaction algorithm and practical Fourier analysis. J.R. Stat. Soc. 20, 361–372 (1958)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Ogura, N., Yamamoto, G., Kobayashi, T., Uchiyama, S.: An Improvement of Key Generation Algorithm for Gentry’s Homomorphic Encryption Scheme. In: Echizen, I., Kunihiro, N., Sasaki, R. (eds.) IWSEC 2010. LNCS, vol. 6434, pp. 70–83. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Rader, C.M.: Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56, 1107–1108 (1968)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Smart, N.P., Vercauteren, F.: Fully Homomorphic SIMD Operations. IACR e-print, 133 (2011)Google Scholar
  12. 12.
    Thomas, L.H.: Using a computer to solve problems in physics. Application of Digital Computers (1963)Google Scholar
  13. 13.
    Trench, W.F.: An algorithm for the inversion of finite Toeplitz matrices. J. SIAM 12, 515–522 (1964)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Peter Scholl
    • 1
  • Nigel P. Smart
    • 1
  1. 1.Dept. Computer ScienceUniversity of BristolBristolUnited Kingdom

Personalised recommendations