## Cryptology ePrint Archive: Report 2014/080

A Full Characterization of Completeness for Two-party Randomized Function Evaluation

Daniel Kraschewski and Hemanta K. Maji and Manoj Prabhakaran and Amit Sahai

Abstract: We settle a long standing open problem which has pursued a full characterization of completeness of (potentially randomized) finite functions for 2-party computation that is secure against active adversaries. Since the first such complete function was discovered [Kilian, FOCS 1988], the question of which finite 2-party functions are complete has been studied extensively, leading to characterization in many special cases. In this work, we completely settle this problem.

We provide a polynomial time algorithm to test whether a 2-party finite secure function evaluation (SFE) functionality (possibly randomized) is complete or not. The main tools in our solution include:

-- A formal linear algebraic notion of redundancy'' in a general 2-party randomized function.

-- A notion of statistically testable games.'' A kind of interactive proof in the information-theoretic setting where both parties are computationally unbounded but differ in their knowledge of a secret.

-- An extension of the (weak) converse of Shannon's channel coding theorem,'' where an adversary can adaptively choose the channel based on it view.

We show that any function f, if complete, can implement any (randomized) circuit C using only O(|C| + k) calls to f, where k is the statistical security parameter. In particular, for any two-party functionality g, this establishes a universal notion of its quantitative cryptographic complexity'' independent of the setup and has close connections to circuit complexity.

Category / Keywords: foundations / Foundations, Secure 2-party Randomized Function Evaluation, Completeness Characterization, Standalone Security, UC Security, Information-theoretic Reduction

Original Publication (with major differences): IACR-EUROCRYPT-2014

Date: received 4 Feb 2014, last revised 5 Feb 2014

Contact author: hemanta maji at gmail com

Available format(s): PDF | BibTeX Citation

Note: This is a FULL VERSION of the paper appearing in Eurocrypt-2014.

Short URL: ia.cr/2014/080

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