Secure Multi-party Computation (MPC) with small population and honest majority has drawn focus specifically due to customization in techniques and resulting efficiency that the constructions can offer. In this work, we investigate a wide range of security notions in the five-party setting, tolerating two active corruptions. Being constant-round, our protocols are best suited for real-time, high latency networks such as the Internet. In a minimal setting of pairwise-private channels, we present efficient instantiations with unanimous abort (where either all honest parties obtain the output or none of them do) and fairness (where the adversary obtains its output only if all honest parties also receive it). With the presence of an additional broadcast channel (known to be necessary), we present a construction with guaranteed output delivery (where any adversarial behaviour cannot prevent the honest parties from receiving the output). The broadcast communication is minimal and independent of circuit size. In terms of performance (communication and run time), our protocols incur minimal overhead over the best known selective abort protocol of Chandran et al. (ACM CCS 2016) while retaining their round complexity. Further, our protocols for fairness and unanimous abort can be extended to n-parties with at most $\sqrt{n}$ corruptions, similar to Chandran et al. Going beyond the most popular honest-majority setting of three parties with one corruption, our results demonstrate feasibility of attaining stronger security notions at an expense not too far from the least desired security of selective abort.

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