Just How Fair is an Unreactive World?

Fitzi, Garay, Maurer, and Ostrovsky (J. Cryptology 2005) showed that in the presence of a dishonest majority, no primitive of cardinality n - 1 is complete for realizing an arbitrary n-party functionality with guaranteed output delivery. In this work, we show that in the presence of n - 1 corrupt parties, no unreactive primitive of cardinality n - 1 is complete for realizing an arbitrary n-party functionality with fairness. We show more generally that for t > n/2, in the presence of t malicious parties, no unreactive primitive of cardinality t is complete for realizing an arbitrary n-party functionality with fairness. We complement this result by noting that (t + 1)-wise fair exchange is complete for realizing an arbitrary n-party functionality with fairness. In order to prove our results, we utilize the primitive of fair coin tossing and the notion of predictability. While this notion has been considered in some form in past works, we come up with a novel and non-trivial framework to employ it, one that readily generalizes from the setting of two parties to multiple parties, and also to the setting of unreactive functionalities.