Efficient private bidding and auctions with an oblivious third party

We describe a novel and efficient protocol for the following problem: <italic>A</italic> wants to buy some good from <italic>B</italic> if the price is less than <italic>a</italic>. <italic>B</italic> would like to sell, but only for more than <italic>b</italic>, and neither of them wants to reveal the secret bounds. Will the deal take place? Our solution uses an oblivious third party <italic>T</italic> who learns no information about <italic>a</italic> or <italic>b</italic>, not even whether <italic>a</italic> > <italic>b</italic>. The protocol needs only a single round of interaction, ensures fairness, and is not based on general circuit evaluation techniques. It uses a novel construction, which combines homomorphic encryption with the φ-hiding assumption and which may be of independent interest. Applications include bargaining between two parties and secure and efficient auctions in the absence of a fully trusted auction service.