Quantum protocols for the millionaire problem with a third party are trivial

Abstract

Recently there were many quantum protocols devoted to solve the millionaire problem and private comparison problem by adding a semi-honest third party. They all require complicated quantum methods, while still leak a non-trivial amount of information to at least one of the parties. But it will be shown here that once the third party is introduced, there are very simple protocols which require quantum key distribution as the only quantum resource, and the amount of information leaked can be made arbitrarily small. Furthermore, even a dishonest third party cannot spoil the protocols. Thus our solutions surpass all existing protocols on both feasibility and security.