Secure multi-party computational geometry

The general secure multi-party computation problem is when multiple parties (say, Alice and Bob) each have private data (respectively, a and b) and seek to compute some function f(a, b) without revealing to each other anything unintended (i.e., anything other than what can be inferred from knowing f(a, b)). It is well known that, in theory, the general secure multi-party computation problem is solvable using circuit evaluation protocols. While this approach is appealing in its generality, the communication complexity of the resulting protocols depend on the size of the circuit that expresses the functionality to be computed. As Goldreich has recently pointed out [6], using the solutions derived from these general results to solve specific problems can be impractical; problem-specific solutions should be developed, for efficiency reasons. This paper is a first step in this direction for the area of computational geometry. We give simple solutions to some specific geometric problems, and in doing so we develop some building blocks that we believe will be useful in the solution of other geometric and combinatorial problems as well.