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 it 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. * This paper is the result of merger of two independent works. One of these works is available online as [Kra13] .
REF extends the study and gives a characterization of all finite two-party functionalities (including randomized functionalities) that are complete against malicious adversaries.
A Full Characterization of Completeness for Two-Party Randomized Function Evaluation
IACR Cryptology ePrint Archive
IACR Cryptology ePrint Archive
Computer Science