A Zero-One Law for Deterministic 2-Party Secure Computation

We use security in the Universal Composition framework as a means to study the “cryptographic complexity” of 2-party secure computation tasks (functionalities). We say that a functionality F reduces to another functionality G if there is a UC-secure protocol for F using ideal access to G. This reduction is a natural and fine-grained way to compare the relative complexities of cryptographic tasks. There are two natural “extremes” of complexity under the reduction: the trivial functionalities, which can be reduced to any other functionality; and the complete functionalities, to which any other functionality can be reduced. In this work we show that under a natural computational assumption (the existence of a protocol for oblivious transfer secure against semi-honest adversaries), there is a zero-one law for the cryptographic complexity of 2-party deterministic functionalities. Namely, every such functionality is either trivial or complete. No other qualitative distinctions exist among functionalities, under this computational assumption. While nearly all previous work classifying multi-party computation functionalities has been restricted to the case of secure function evaluation, our results are the first to consider completeness of arbitrary reactive functionalities, which receive input and give output repeatedly throughout several rounds of interaction. One important technical contribution in this work is to initiate the comprehensive study of the cryptographic properties of reactive functionalities. Wemodel these functionalities as finite automata and develop an automata-theoretic methodology for classifying and studying their cryptographic properties. Consequently, we completely characterize the reactive behaviors that lead to cryptographic non-triviality. ∗Work supported by NSF grants CNS 07-16626 and CNS 07-47027. †Department of Computer Science, University of Illinois, Urbana-Champaign. {hmaji2,mmp}@uiuc.edu. ‡Department of Computer Science, University of Montana. [email protected].