Asymptotics of quantum contract signing

Distance contract signing has been an important topic in cryptography, relevant to numerous applications, including stock market [1]. It involves two parties, called Alice and Bob, who want to exchange their commitment to a contract. The fundamental problem is that one party has to go first in sending their commitment, which gives the other an advantage, and one would like to design a protocol which is both fair and viable. A protocol is fair if neither party can get a commitment if it does not give it at the same time. An unfair protocol would, for instance, give a trader a right to trade future or option if it is favorable to him, without having to pay if things do not turn advantageously. A protocol is viable if it enables signing parties to get each other commitments provided they both act honestly. It can be shown (see [3]) that it is impossible to design a fair and viable contract signing protocol, without involving a third, trusted party (called Trent in cryptography). If the third party is involved, then Alice and Bob could, for instance, send their commitments to Trent, who would send them back in a way that ensures fairness and viability. However, it is desirable to involve Trent only if necessary. A protocol is called optimistic, if the third, trusted party, is involved only when one party is cheating or the communication is interrupted. In optimistic protocols, Alice and Bob exchange messages, so that in the end both parties will end up with signed contract. However, if there is a disruption or evidence of cheating, the parties have an option to invoke Trent, who would then bind the contract, assuring fairness.