Sublinear-Communication Secure Multiparty Computation Does Not Require FHE.

Secure computation enables mutually distrusting parties to jointly compute a function on their secret inputs, while revealing nothing beyond the function output. A long-running challenge is understanding the required communication complexity of such protocols—in particular, when communication can be sublinear in the circuit representation size of the desired function. Significant advances have been made affirmatively answering this question within the two-party setting, based on a variety of structures and hardness assumptions. In contrast, in the multi-party setting, only one general approach is known: using Fully Homomorphic Encryption (FHE). This remains the state of affairs even for just three parties, with two corruptions. We present a framework for achieving secure sublinear-communication $$(N+1)$$ -party computation, building from a particular form of Function Secret Sharing for only N parties. In turn, we demonstrate implications to sublinear secure computation for various function classes in the 3-party and 5-party settings based on an assortment of assumptions not known to imply FHE.