Strongly Multiplicative and 3-Multiplicative Linear Secret Sharing Schemes

  • Zhifang Zhang
  • Mulan Liu
  • Yeow Meng Chee
  • San Ling
  • Huaxiong Wang
Conference paper

DOI: 10.1007/978-3-540-89255-7_2

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5350)
Cite this paper as:
Zhang Z., Liu M., Chee Y.M., Ling S., Wang H. (2008) Strongly Multiplicative and 3-Multiplicative Linear Secret Sharing Schemes. In: Pieprzyk J. (eds) Advances in Cryptology - ASIACRYPT 2008. ASIACRYPT 2008. Lecture Notes in Computer Science, vol 5350. Springer, Berlin, Heidelberg

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 
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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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