Hiccups on the road to privacy-preserving linear programming

Linear programming is one of maths' greatest contributions to industry. There are many places where linear programming could be beneficially applied across more than one company, but there is a roadblock. Companies have secrets. The data needed for joint optimization may need to be kept private either through concerns about leaking competitively sensitive data, or due to privacy legislation. Recent research has tackled the problem of privacy-preserving linear programming. One appealing group of approaches uses a 'disguising' transformation to allow one party to perform the joint optimization without seeing the secret data of the other parties. These approaches are very appealing from the point of view of simplicity, efficiency, and flexibility, but we show here that all of the existing transformations have a critical flaw.