We study the round complexity of multiparty computation with fairness
and guaranteed output delivery, assuming existence of an honest majority. We demonstrate a new lower bound and a matching
upper bound. Our lower bound rules out any two-round
fair protocols in the standalone model, even when the parties are given access to a common reference string (CRS). The lower bound follows by a reduction to the impossibility result of virtual black box obfuscation of arbitrary circuits.
Then we demonstrate a three-round protocol with guarantee of output delivery, which in general is harder than achieving fairness (since the latter allows the adversary to force a fair abort). We develop a new construction of a threshold fully homomorphic encryption scheme, with a new property that we call ``flexible'' ciphertexts. Roughly, our threshold encryption scheme allows parties to adapt flexible ciphertexts to the public keys of the non-aborting parties, which provides a way of handling aborts without adding any communication.
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