In this work, we present novel protocols over rings for semi-honest secure three-party and malicious four-party computation with one corruption. Our protocols are optimized for heterogeneous network settings. $P_1$ and $P_2$ do not need to communicate with each other, while $P_1$ and $P_3$ communicate only in the offline phase in a single communication round. Our four-party protocol additionally requires $P_4$ to only communicate with $P_1$ in the offline phase in a single communication round. Thus, only $P_2$ and $P_3$ need to share a low latency, high bandwidth channel to achieve fast runtimes. We list several applications where this property can be useful. For instance, if $P_1$ and $P_2$ want to perform a two-party computation but share a weak network link they can utilize an auxiliary $P_3$ with a strong network link to $P_2$ to accelerate the secure computation. Our three-party protocol requires 1 domain element of global communication in the offline phase and 2 elements of global communication in the online phase. Our four-party protocol requires 2 elements of global communication in the offline phase and 3 elements of global communication in the online phase. This protocol can provide fairness and guaranteed output delivery. The best previous four-party malicious protocols with one corruption require 6 elements of global communication. We also show different ways of how to transform our protocols into homogeneous protocols where parties communicate equally on all links at the same global communication complexity.

翻译：本文提出针对半诚实安全三方计算与单方腐败恶意四方计算的新型环上协议。我们的协议针对异构网络环境进行了优化。$P_1$与$P_2$无需相互通信，而$P_1$与$P_3$仅在离线阶段通过单轮通信进行交互。四方协议额外要求$P_4$仅在离线阶段与$P_1$进行单轮通信。因此，仅需$P_2$与$P_3$共享低延迟、高带宽信道即可实现快速运行。我们列举了该特性可发挥作用的若干应用场景。例如，若$P_1$与$P_2$需进行两方计算但网络链路较弱，可借助与$P_2$具有强网络链路的辅助方$P_3$来加速安全计算。三方协议离线阶段需1个全局通信域元素，在线阶段需2个元素；四方协议离线阶段需2个全局通信域元素，在线阶段需3个元素。该协议可实现公平性与保证输出交付。此前最优的单方腐败四方恶意协议需6个全局通信元素。我们还展示了如何将协议转化为所有链路上通信量均衡的同构协议，且保持相同全局通信复杂度。