Complete fairness in secure two-party computation

In the setting of secure two-party computation, two mutually distrusting parties wish to compute some function of their inputs while preserving, to the extent possible, various security properties such as privacy, correctness, and more. One desirable property is fairness, which guarantees that if either party receives its output, then the other party does too. Cleve (STOC 1986) showed that complete fairness cannot be achieved in general in the two-party setting; specifically, he showed (essentially) that it is impossible to compute Boolean XOR with complete fairness. Since his work, the accepted folklore has been that nothing non-trivial can be computed with complete fairness, and the question of complete fairness in secure two-partyr computation has been treated as closed since the late '80s. In this paper, we demonstrate that this widely held folklore belief is false by showing completely-fair secure protocols for various non-trivial two-party functions including Boolean AND/OR as well as Yao's "millionaires' problem". Surprisingly, we show that it is even possible to construct completely-fair protocols for certain functions containing an "embedded XOR", although in this case we also prove a lower-bound showing that a super-logarithmic number of rounds are necessary. Our results demonstrate that the question of completely-fair secure computation without an honest majority is far from closed.