A full characterization of completeness for two-party randomized function evaluation

We settle a long standing open problem which has pursued a full characterization of completeness of (potentially randomized) finite functions for 2-party computation that is secure against active adversaries. Since the first such complete function was discovered [Kilian, FOCS 1988], the question of which finite 2-party functions are complete has been studied extensively, leading to characterization in many special cases. In this work, we completely settle this problem. We provide a polynomial time algorithm to test whether a 2-party finite secure function evaluation (SFE) functionality (possibly randomized) is complete or not. The main tools in our solution include: - A formal linear algebraic notion of redundancy in a general 2-party randomized function. - A notion of statistically testable games. A kind of interactive proof in the information-theoretic setting where both parties are computationally unbounded but differ in their knowledge of a secret. - An extension of the (weak) converse of Shannon's channel coding theorem, where an adversary can adaptively choose the channel based on its view. We show that any function f, if complete, can implement any (randomized) circuit C using only O(|C| + κ) calls to f, where κ is the statistical security parameter. In particular, for any two-party functionality g, this establishes a universal notion of its quantitative "cryptographic complexity" independent of the setup and has close connections to circuit complexity.