Round complexity is a central measure of efficiency, and characterizing the round complexity of various cryptographic tasks is of both theoretical and practical importance. We show here a universally-composable (UC) protocol (in the common reference string model) for two-party computation of any functionality, where
parties receive output, using only two rounds. (This assumes honest parties are allowed to transmit messages simultaneously in any given round; we obtain a three-round protocol when parties are required to alternate messages.) Our results match the obvious lower bounds for the round complexity of secure two-party computation under any reasonable definition of security, regardless of what setup is used. Thus, our results establish that secure two-party computation can be obtained under a commonly-used setup assumption with
security (i.e., security under general composition) in a
number of rounds.
To give but one example of the power of our general result, we observe that as an almost immediate corollary we obtain a two-round UC blind signature scheme, matching a result by Fischlin at Crypto 2006 (though, in contrast to Fischlin, we use specific number-theoretic assumptions).