Fair and optimistic quantum contract signing

Abstract

We present a fair and optimistic [7,8] quantum contract signing protocol between two clients that requires no communication with the third trusted party during the exchange phase. We discuss its fairness and show that it is possible to design such a protocol for which the probability of a dishonest client to cheat becomes negligible, and scales as N −1/2 , where N is the number of messages exchanged between the clients. Our protocol is not based on the exchange of signed messages: its fairness is based on the laws of quantum mechanics. Thus, it is abuse-free [9], and the clients do not have to generate new keys for each message during the Exchange phase. We discuss a real-life scenario when the measurement errors and qubit state corruption due to noisy channels occur and argue that for real, good enough measurement apparatus and transmission channels, our protocol would still be fair. Our protocol could be implemented by today's technology, as it requires in essence the same type of apparatus as the one needed for BB84 cryptographic protocol [12]. Finally, we briefly discuss two alternative versions of the protocol, one that uses only two states (based on B92 protocol [24]) and the other that uses entangled pairs (based on [20]), and show that it is possible to generalize our protocol to an arbitrary number of clients.

Secure N-dimensional simultaneous dense coding and applications

“…Yet, the protocol is fair [39]: throughout the execution, one client is only slightly privileged over the other, the difference being smaller with the increase of s and could be made arbitrarily small for big enough s. The argument here is exactly the same as the one presented for the fairness of the quantum contract signing protocol [37].…”

“…The situation is similar to the one presented in the contract signing problem [34]. The solution, similar to the one proposed for quantum contract signing [37], is to perform the unlocking operation on quN its A 1 and A 2 (i.e., arrays of qubits A 1,1 . .…”

“…In the case of quantum contract signing [37] the role of message and control qubits was given to the same N qubits sent by a trusted party (in our case Alice) to clients (in our case Bob and Charlie), while in the case of simultaneous dense coding the two roles are given to separate sets of qubits. This will introduce a slight change in the expressions for the probabilities involved in calculation, but this change is minor, conceptually straightforward to calculate and does not affect the main result of protocol's fairness.…”

“…The fairness condition can be even straightened, as was done in [37], by requiring the negligible probability to cheat: the probability that one client has 2αn right bits, while the other does not. This quantity may be quite relevant in various possible scenarios, like in the case of signing contracts for buying and selling the goods on the market (see [37]).…”

“…The fairness condition can be even straightened, as was done in [37], by requiring the negligible probability to cheat: the probability that one client has 2αn right bits, while the other does not. This quantity may be quite relevant in various possible scenarios, like in the case of signing contracts for buying and selling the goods on the market (see [37]). While for a fixed value of α the probability to cheat may be as high as 1/4, if it is unknown to the clients and chosen randomly by Alice from a certain interval I α ⊂ (1/2, 1), the probability to cheat can be made as small as needed, for big enough n (see [37]).…”