Secure multiparty computation of solid geometric problems and their applications

Secure multiparty computation is an area of recent research in the international cryptographic community, and secure computational geometry is an essential field of secure multiparty computation. Most of the existing studies of secure multiparty computational geometric problems have focused on plane geometry, while very few have addressed solid geometry. Because solid geometry is an integral part of geometry and describes the real world better than plane geometry, research on secure computational solid geometry is appealing. Motivated by an interesting application, we first examine the problem of the secure multiparty computation of a tetrahedron, propose a solution, and prove that the solution is private using an accepted simulation paradigm. Using the solution to the tetrahedron problem as a building block, we further solve the secure multiparty computation of three other solid geometric problems, including the relationship between a point and a plane, the relationship between a line and a plane, and the relationship between two planes. We also demonstrate that the solutions to these problems are private. We analyze the computational and communication complexities of these solutions and show that the computational complexities are near or equal to the problems' minimum theoretical computational complexity and that the communication complexities are equal to the minimum theoretical communication complexity. Thus, these solutions are optimal. Finally, we show an interesting application of secure computational solid geometry. (C) 2014 Elsevier Inc. All rights reserved.