Strongly Multiplicative and 3-Multiplicative Linear Secret Sharing Schemes

  • Zhifang Zhang
  • Mulan Liu
  • Yeow Meng Chee
  • San Ling
  • Huaxiong Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5350)

Abstract

Strongly multiplicative linear secret sharing schemes (LSSS) have been a powerful tool for constructing secure multi-party computation protocols. However, it remains open whether or not there exist efficient constructions of strongly multiplicative LSSS from general LSSS. In this paper, we propose the new concept of 3-multiplicative LSSS, and establish its relationship with strongly multiplicative LSSS. More precisely, we show that any 3-multiplicative LSSS is a strongly multiplicative LSSS, but the converse is not true; and that any strongly multiplicative LSSS can be efficiently converted into a 3-multiplicative LSSS. Furthermore, we apply 3-multiplicative LSSS to the computation of unbounded fan-in multiplication, which reduces its round complexity to four (from five of the previous protocol based on multiplicative LSSS). We also give two constructions of 3-multiplicative LSSS from Reed-Muller codes and algebraic geometric codes. We believe that the construction and verification of 3-multiplicative LSSS are easier than those of strongly multiplicative LSSS. This presents a step forward in settling the open problem of efficient constructions of strongly multiplicative LSSS from general LSSS.

Keywords

monotone span program secure multi-party computation strongly multiplicative linear secret sharing scheme 

References

  1. 1.
    Bar-Ilan, J., Beaver, D.: Non-cryptographic fault-tolerant computing in constant number of rounds of interaction. In: PODC 1989, pp. 201–209 (1989)Google Scholar
  2. 2.
    Beimel, A.: Secure schemes for secret sharing and key distribution. PhD thesis, Technion - Israel Institute of Technology (1996)Google Scholar
  3. 3.
    Chen, H., Cramer, R.: Algebraic geometric secret sharing schemes and secure multi-party computations over small fields. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 521–536. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Cramer, R., Kiltz, E., Padró, C.: A note on secure computation of the Moore-Penrose pseudoinverse and its spplication to secure linear algebra. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 613–630. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Chen, H., Cramer, R., de Haan, R., Cascudo Pueyo, I.: Strongly multiplicative ramp schemes from high degree rational points on curves. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 451–470. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Cramer, R., Damgård, I., Maurer, U.: General secure multi-party computation from any linear secret-sharing scheme. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 316–334. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Cramer, R., Daza, V., Gracia, I., Urroz, J., Leander, G., Martí-Farré, J., Padró, C.: On codes, matroids and secure multi-party computation from linear secret sharing schemes. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 327–343. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Fehr, S.: Efficient construction of the dual span program. Master Thesis, the Swiss Federal Institute of Technology (ETH) Zürich (1999), http://homepages.cwi.nl/~fehr/publications.html
  9. 9.
    Goldreich, O., Micali, S., Wigderson, A.: How to play ANY mental game. In: STOC 1987, pp. 218–219 (1987)Google Scholar
  10. 10.
    Karchmer, M., Wigderson, A.: On span programs. In: Proc. 8th Ann. Symp. Structure in Complexity Theory, pp. 102–111 (1993)Google Scholar
  11. 11.
    Käsper, E., Nikov, V., Nikova, S.: Strongly multiplicative hierarchical threshold secret sharing. In: 2nd International Conference on Information Theoretic Security - ICITS 2007. LNCS (to appear, 2007)Google Scholar
  12. 12.
    Liu, M., Xiao, L., Zhang, Z.: Multiplicative linear secret sharing schemes based on connectivity of graphs. IEEE Transactions on Information Theory 53(11), 3973–3978 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Massey, J.L.: Minimal codewords and secret sharing. In: Proc. 6th Joint Swedish-Russian Workshop on Information Theory, pp. 276–279 (1993)Google Scholar
  14. 14.
    Nikov, V., Nikova, S., Preneel, B.: On multiplicative linear secret sharing schemes. In: Johansson, T., Maitra, S. (eds.) INDOCRYPT 2003. LNCS, vol. 2904, pp. 135–147. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    van Lint, J.H.: Introduction to coding theory, 3rd edn. Graduate Texts in Mathematics, vol. 86. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  16. 16.
    Yao, A.: Protocols for secure computation. In: FOCS 1982, pp. 160–164 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Zhifang Zhang
    • 1
  • Mulan Liu
    • 1
  • Yeow Meng Chee
    • 2
  • San Ling
    • 2
  • Huaxiong Wang
    • 2
    • 3
  1. 1.Key Laboratory of Mathematics MechanizationAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore
  3. 3.Centre for Advanced Computing - Algorithms and Cryptography Department of ComputingMacquarie UniversityAustralia

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