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.
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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. https://doi.org/10.1007/978-3-540-89255-7_2
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DOI: https://doi.org/10.1007/978-3-540-89255-7_2
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