On Privacy Preserving Convex Hull

Computing convex hull for a given set of points is one of the most explored problems in the area of computational geometry (CG). If the set of points is distributed among a set of parties who jointly wish to compute the convex hull, each party can send his points to every other party, and can then locally compute the hull using any of the existing algorithms in CG. However such an approach does not work if the parties wish to compute the convex hull securely, i.e., no party wishes to reveal any of his input points to any other party apart from those that are part of the answer. The problem of secure computation of convex hull for two parties was first introduced by Du and Atallah (NSPW ´01). The first solution to the problem was given by Wang et. al(ARES ´08). However, the proposed solution was based on well known algorithms for computing convex hull in CG which are proven to be sub-optimal. We propose a new solution for secure computation of convex hull with a considerable improvement in computational complexity. We further show how to extend our two-party protocol for the case of any number of parties.