KSII Transactions on Internet and Information Systems (TIIS)

Korean Society for Internet Information (한국인터넷정보학회)
권호 표지
DOI QR코드

DOI QR Code

An Efficient Somewhat HE scheme over Integers and Its Variation

  • Yang, Haomiao (School of Computer Science & Engineering, UESTC) ;
  • Kim, Hyunsung (Dept. of Cyber Security, Kyungil University) ;
  • Tang, Dianhua (Science & Technology on Communication Security Laboratory) ;
  • Li, Hongwei (School of Computer Science & Engineering, UESTC)
  • Received : 2013.04.15
  • Accepted : 2013.09.21
  • Published : 2013.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In 2010, Dijk et al. demonstrated a simple somewhat homomorphic encryption (HE) scheme over the integers of which this simplicity came at the cost of a public key size in $\tilde{O}({\lambda}^{10})$. Although in 2011 Coron et al. reduced the public key size to $\tilde{O}({\lambda}^7)$, it is still too large for practical applications, especially for the cloud computing. In this paper, we propose a new form of somewhat HE scheme to reduce further the public key size and a variation of the scheme to optimize the ciphertext size. First of all, we propose a new somewhat HE scheme which is built on the hardness of the approximate greatest common divisor (GCD) problem of two integers, where the public key size in the scheme is reduced to $\tilde{O}({\lambda}^3)$. Furthermore, we can reduce the length of the ciphertext of the new somewhat HE scheme by applying the modular reduction technique. Additionally, we give simulation results for evaluating ability of the proposed scheme.

Keywords

References

  1. J. Shi, R. Zhang, Y. Liu, and Y. Zhang, "Prisense: privacy-preserving data aggregation in people-centric urban sensing systems," in Proc. of INFOCOM, 2010, pp. 1-9.
  2. H. Li, X. Lin, H. Yang, X. Liang, R. Lu, and X. Shen, "EPPDR: An Efficient Privacy-Preserving Demand Response Scheme with Adaptive Key Evolution in Smart Grid, " IEEE Transactions on Parallel and Distributed Systems, 2013, to appear.
  3. R. Lu, X. Liang, X. Li, X. Lin, and X. Shen, "Eppa: An efficient and privacy preserving aggregation scheme for secure smart grid communications," IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 9, pp. 1621-1631, 2012. https://doi.org/10.1109/TPDS.2012.86
  4. R. Gennaro, C. Gentry, and B. Parno, "Non-interactive verifiable computing: Outsourcing computation to untrusted workers," in Proc. of Advances in Cryptology - CRYPTO 2010, Springer Berlin / Heidelberg, LNCS 6223, 2010, pp. 465-482.
  5. K.-M. Chung, Y. T. Kalai and S. P. Vadhan, "Improved delegation of computation using fully homomorphic encryption," in Proc. of Advances in Cryptology - CRYPTO 2010, Springer Berlin / Heidelberg, LNCS 6223, 2010, pp. 483-501.
  6. Y. Gahi, M. Guennoun and K. El-Khatib, "A secure database system using homomorphic encryption schemes" in Proc. of The Third International Conference on Advances in Databases, Knowledge, and Data Applications. 2011, pp. 54-58. http://www.thinkmind.org/index.php?view=article&articleid=dbkda_2011_3_20_30074
  7. R. L. Rivest, L. M.Adleman, and M. L.Dertouzos, "On data banks and privacy homomorphisms," in Proc. of Foundations of Sec. Comp., 1978, pp.169-180. http://www.citeulike.org/user/deitosrafael/article/3877157
  8. D. Boneh, E.-J Goh, and K. Nissim, "Evaluating 2-DNF Formulas on Ciphertexts," in Proc. of Theory of Cryptography Conference, Springer Berlin / Heidelberg, LNCS 3378, 2005, pp.325-341.
  9. C. Gentry, "Fully homomorphic encryption using ideal lattices," STOC 2009, ACM, 2009, pp.169-178.
  10. D. Stehlé, and R. Steinfeld, "Faster Fully Homomorphic Encryption," in Proc. of Advances in Cryptology - ASIACRYPT 2010, Springer Berlin / Heidelberg, LNCS 6477, 2010, pp.377-394.
  11. Z. Brakerski, C. Gentry, and V. Vaikuntanathan, "Fully Homomorphic Encryption without Bootstrapping," http://eprint.iacr.org/2011/
  12. C. Gentry, "A fully homomorphic encryption scheme," Stanford University, PhD Thesis, 2009.
  13. Z. Brakerski and V. Vaikuntanathan, "Efficient fully homomorphic encryption from (standard) LWE," in Proc. of IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011, pp. 97-106.
  14. C. Gentry and S. Halevi, "Fully homomorphic encryption without squashing using depth-3 arithmetic circuits," in Proc. of IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011, pp. 107-109.
  15. M. Dijk, C. Gentry, S. Halevi, and V. Vaikuntanathan, "Fully Homomorphic Encryption over the Integers," in Proc. of Advances in Cryptology - EUROCRYPT 2010, Springer Berlin / Heidelberg, LNCS 6110, 2010, pp. 24-43.
  16. J.-S. Coron, A. Mandal, D. Naccache, and M. Tibouchi, "Fully Homomorphic Encryption over the Integers with Shorter Public Keys," in Proc. of Advances in Cryptology - CRYPTO 2011, Springer Berlin / Heidelberg, LNCS 6841, 2011, pp.487-504.
  17. D. E. Knuth. Asymptotic Representations, volume 1 of The Art of Computer Programming, Addison-Wesley, 3rd edition, 1997. http://www.amazon.com/Art-Computer-Programming-Volume-Fundamental/dp/0201896834
  18. N. Howgrave-Graham, "Approximate integer common divisors," in Proc. of CaLC' 01, Springer, LNCS 2146, 2001, pp.51-66. http://dl.acm.org/citation.cfm?id=753508
  19. D. E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming, Addison-Wesley, 3rd edition, 1997. http://www.amazon.com/Art-Computer-Programming-Volume-Seminumerical/dp/0201896842
  20. H. Yang, D. Tang, Q. Xia, and X. Wang, "A New Somewhat Homomorphic Encryption Scheme over Integers," in Proc. of 2012 International Conference on Computer Distributed Control and Intelligent Environmental Monitoring, CDCIEM 2012, 2012, pp. 61-64.
  21. V. Shoup, NTL: A Library for doing Number Theory. http://shoup.net/ntl/