Fully Homomorphic Encryption over the Integers

  • Marten van Dijk
  • Craig Gentry
  • Shai Halevi
  • Vinod Vaikuntanathan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6110)

Abstract

We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct a fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of our scheme is the conceptual simplicity.

We reduce the security of our scheme to finding an approximate integer gcd – i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer. We investigate the hardness of this task, building on earlier work of Howgrave-Graham.

References

  1. 1.
    Alexi, W., Chor, B., Goldreich, O., Schnorr, C.-P.: Rsa and rabin functions: Certain parts are as hard as the whole. SIAM J. Comput. 17(2), 194–209 (1988)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cachin, C., Micali, S., Stadler, M.: Computationally private information retrieval with polylogarithmic communication. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 402–414. Springer, Heidelberg (1999)Google Scholar
  4. 4.
    Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Cryptology 10(4), 233–260 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009), http://crypto.stanford.edu/craig
  6. 6.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC 2009, pp. 169–178. ACM, New York (2009)CrossRefGoogle Scholar
  7. 7.
    Goldwasser, S., Micali, S.: Probabilistic encryption. Journal of Computer and System Sciences 28(2), 270–299 (1984)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Howgrave-Graham, N.: Approximate integer common divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Ishai, Y., Paskin, A.: Evaluating branching programs on encrypted data. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 575–594. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Karp, R.M., Ramachandran, V.: A Survey of Parallel Algorithms for Shared-Memory Machines. Technical Report CSD-88-408, UC Berkeley (1988)Google Scholar
  12. 12.
    Knuth, D.E.: Seminumerical Algorithms, 3rd edn. The Art of Computer Programming, vol. 2. Addison-Wesley, Reading (1997)Google Scholar
  13. 13.
    Lagarias, J.C.: The computational complexity of simultaneous diophantine approximation problems. SIAM J. Comput. 14(1), 196–209 (1985)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lenstra, A.K.: Factoring multivariate polynomials over algebraic number fields. SIAM J. Comput. 16(3), 591–598 (1987)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lindell, Y., Pinkas, B.: A proof of security of yao’s protocol for two-party computation. J. Cryptology 22(2) (2009)Google Scholar
  16. 16.
    Nguyen, P.Q., Shparlinski, I.: On the insecurity of a server-aided RSA protocol. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 21–35. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Nguyen, P.Q., Stern, J.: Adapting density attacks to low-weight knapsacks. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 41–58. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Regev, O.: New lattice-based cryptographic constructions. JACM 51(6), 899–942 (2004)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rivest, R., Adleman, L., Dertouzos, M.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, pp. 169–177. Academic Press, London (1978)Google Scholar
  21. 21.
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. Cryptology ePrint Archive, Report 2009/616 (2009), http://eprint.iacr.org/2009/616
  22. 22.
    Yao, A.C.: Protocols for secure computations. In: 23rd Annual Symposium on Foundations of Computer Science – FOCS 1982, pp. 160–164. IEEE, Los Alamitos (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marten van Dijk
    • 1
  • Craig Gentry
    • 2
  • Shai Halevi
    • 2
  • Vinod Vaikuntanathan
    • 2
  1. 1.MIT CSAIL 
  2. 2.IBM Research 

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