Secure Two-Party Computation (STC) allows two mutually untrusting parties to securely evaluate a function on their private inputs. While tremendous progress has been made towards reducing processing overheads, STC still incurs significant communication overhead that is in fact prohibitive when no high-speed network connection is available, e.g., when applications are run over a cellular network. In this paper, we consider the fundamental problem of securely computing a minimum and its argument, which is a basic building block in a wide range of applications that have been proposed as STCs, e.g., Nearest Neighbor Search, Auctions, and Biometric Matchings. We first comprehensively analyze and compare the communication overhead of implementations of the three major STC concepts, i.e., Yao's Garbled Circuits, the Goldreich-Micali-Wigderson protocol, and Homomorphic Encryption. We then propose an algorithm for securely computing minima in the semi-honest model that, compared to current state-of-the-art, reduces communication overheads by 18 % to 98 %. Lower communication overheads result in faster runtimes in constrained networks and lower direct costs for users.
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