On the Structure of Unconditional UC Hybrid Protocols

We study the problem of secure two-party computation in the presence of a trusted setup. If there is an unconditionally UC-secure protocol for f that makes use of calls to an ideal g, then we say that f reduces to g (and write f v g). Some g are complete in the sense that all functions reduce to g. However, almost nothing is known about the power of an incomplete g in this setting. We shed light on this gap by showing a characterization of f v g for incomplete g. Very roughly speaking, we show that f reduces to g if and only if it does so by the simplest possible protocol: one that makes a single call to ideal g and uses no further communication. Furthermore, such simple protocols can be characterized by a natural combinatorial condition on f and g. Looking more closely, our characterization applies only to a very wide class of f , and only for protocols that are deterministic or logarithmic-round. However, we give concrete examples showing that both of these limitations are inherent to the characterization itself. Functions not covered by our characterization exhibit qualitatively different properties. Likewise, randomized, superlogarithmic-round protocols are qualitatively more powerful than deterministic or logarithmic-round ones.