  Secure multi-party computation (MPC) allows a set of parties to compute a
function jointly while keeping their inputs private. Compared with the MPC
based on garbled circuits,some recent research results show that MPC based on
secret sharing (SS) works at a very high speed. Moreover, SS-based MPC can be
easily vectorized and achieve higher throughput. In SS-based MPC, however, we
need many communication rounds for computing concrete protocols like equality
check, less-than comparison, etc. This property is not suited for large-latency
environments like the Internet (or WAN). In this paper, we construct
semi-honest secure communication-efficient two-party protocols. The core
technique is Beaver triple extension, which is a new tool for treating
multi-fan-in gates, and we also show how to use it efficiently. We mainly focus
on reducing the number of communication rounds, and our protocols also succeed
in reducing the number of communication bits (in most cases). As an example, we
propose a less-than comparison protocol (under practical parameters) with three
communication rounds. Moreover, the number of communication bits is also
$38.4\%$ fewer. As a result, total online execution time is $56.1\%$ shorter
than the previous work adopting the same settings. Although the computation
costs of our protocols are more expensive than those of previous work, we
confirm via experiments that such a disadvantage has small effects on the whole
online performance in the typical WAN environments.
