Secure and Private Collaborative Linear Programming

The growth of the Internet has created tremendous opportunities for online collaborations. These often involve collaborative optimizations where the two parties are, for example, jointly minimizing costs without violating their own particular constraints (e.g., one party may have too much inventory, another too little inventory but too much production capacity, etc). Many of these optimizations can be formulated as linear programming problems, or, rather, as collaborative linear programming, in which two parties need to jointly optimize based on their own private inputs. It is often important to have online collaboration techniques and protocols that carry this out without either party revealing to the other anything about their own private inputs to the optimization (other than, unavoidably, what can be deduced from the collaboratively computed optimal solution). For example, two organizations who jointly invest in a project may want to minimize some linear objective function while satisfying both organizations' private and confidential constraints. Constraints are usually private when they reveal too much about the organizations' financial health, its future business strategy, etc. Linear programming problems have been widely studied in the literature. However, the existing solutions (e.g., the simplex method) do not extend to the above-mentioned framework in which the linear constraints are shared by the two parties, who do not want to disclose their own to the other party. In this paper, we give an efficient protocol for solving linear programming problems in the honest-but-curious model, such that neither party reveals anything about their private input to the other party (other than what can be inferred from the result). The amount of communication and computation done by our protocol is proportional to the time complexity of the simplex method, a widely used linear programming algorithm. We also provide a practical solution that prevents certain malicious behavior of the participants. The use of the known general circuit-simulation solutions to secure function evaluation is unacceptable for the simplex method, as it implies an exponential size circuit