Paper 2016/252

The Exact Round Complexity of Secure Computation

Sanjam Garg, Pratyay Mukherjee, Omkant Pandey, and Antigoni Polychroniadou

Abstract

We revisit the exact round complexity of secure computation in the multi-party and two-party settings. For the special case of two-parties without a simultaneous message exchange channel, this question has been extensively studied and resolved. In particular, Katz and Ostrovsky (CRYPTO '04) proved that 5 rounds are necessary and sufficient for securely realizing every two-party functionality where both parties receive the output. However, the exact round complexity of general multi-party computation, as well as two-party computation with a simultaneous message exchange channel, is not very well understood. These questions are intimately connected to the round complexity of non-malleable commitments. Indeed, the exact relationship between the round complexities of non-malleable commitments and secure multi-party computation has also not been explored. In this work, we revisit these questions and obtain several new results. First, we establish the following main results. Suppose that there exists a k-round non-malleable commitment scheme, and let k' = max(4, k + 1); then, – (Two-party setting with simultaneous message transmission): there exists a k'-round protocol for securely realizing every two-party functionality; – (Multi-party setting):there exists a k'-round protocol for securely realizing the multi-party coin-flipping functionality. As a corollary of the above results, by instantiating them with existing non-malleable commitment protocols (from the literature), we establish that four rounds are both necessary and sufficient for both the results above. Furthermore, we establish that, for every multi-party functionality five rounds are sufficient. We actually obtain a variety of results offering trade-offs between rounds and the cryptographic assumptions used, depending upon the particular instantiations of underlying protocols.

Note: Revised version according to the Ph.D. thesis of Antigoni Polychroniadou. More specifically, the requirement of 3-robust non-malleable commitments is incorporated in this version.