Full Characterization of Completeness for Two-party Randomized Function Evaluation

We close a long line of work that has pursued a full characterization of completeness of (possibly randomized) finite functions for 2-party computation that is secure against active adversaries. The first such function was discovered almost quarter of a century back (Kilian, FOCS 1988). Since then the question of which all finite 2-party functions are complete has been studied extensively, leading to characterizations in many special cases. In this work, we answer this problem in full. The main tools in our solution include: – a linear-algebraic characterization of redundancy in a 2-party function, – the notion of statistical testability (a version of interactive proofs in the informationtheoretic setting) and a result that evaluation of a 2-party function is statistically testable if and only if the function is redundancy free, and – an extension to the (weak) converse of Shannon’s channel coding theorem. Our main construction also gives a generalization of a line of work on obtaining protocols secure against active adversaries from protocols secure against passive adversaries in a black-box manner. ∗University of California, Los Angeles. Supported by NSF CI Postdoctoral Fellowship. [email protected]. †University of Illinois, Urbana-Champaign. Supported by NSF grant CNS 07-47027. [email protected].