  Strongly multiplicative linear secret sharing schemes (LSSS) have been a
powerful tool for constructing secure multiparty 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 a 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 strongly
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.
