Geometry of Secure Two-party Computation.

What is the round and communication complexity of secure computation? The seminal results of Chor-Kushilevitz-Beaver (STOC-1989, FOCS-1989, DIMACS-1989) answer this question for computations with deterministic output. However, this question has remained unanswered for computations with randomized output. Our work answers this question for two-party secure function evaluation functionalities. We introduce a geometric encoding of all candidate secure protocols for a given computation as points in a high-dimensional space. The following results follow by analyzing the properties of these sets of points. 1) It is decidable to determine if a given computation has a secure protocol within round or communication constraints. 2) We construct one such protocol if it exists. 3) Otherwise, we present an obstruction to achieving security. Our technical contributions imply new information complexity bounds for secure computation.