Completeness Theorems with Constructive Proofs for Finite Deterministic 2-Party Functions

In this paper we present simple but comprehensive combinatorial criteria for completeness of finite deterministic 2-party functions with respect to information-theoretic security. We give a general protocol construction for efficient and statistically secure reduction of oblivious transfer to any finite deterministic 2-party function that fulfills our criteria. For the resulting protocols we prove universal composability. Our results are tight in the sense that our criteria still are necessary for any finite deterministic 2-party function to allow for implementation of oblivious transfer with statistical privacy and correctness.

We unify and generalize results of Joe Kilian (1991, 2000) in two ways. Firstly, we show that his completeness criteria also hold in the UC framework. Secondly, what is our main contribution, our criteria also cover a wide class of primitives that are not subject of previous criteria. We show that there are non-trivial examples of finite deterministic 2-party functions that are neither symmetric nor asymmetric and therefore have not been covered by existing completeness criteria so far.

As a corollary of our work, every finite deterministic 2-party function is either complete or can be considered equivalent to a non-complete symmetric 2-party function—this assertion holds true with respect to active adversaries as well as passive adversaries. Thereby known results on non-complete symmetric 2-party functions are strengthened.