Multi-party contract signing protocols specify how a number of signers can cooperate in achieving a fully signed contract, even in the presence of dishonest signers. This problem has been studied in different settings, yielding solutions of varying complexity. Here we assume the presence of a trusted third party that will be contacted only in case of a conflict, asynchronous communication, and a total ordering of the protocol steps. Our goal is to develop a lower bound on the number of messages in such a protocol. Using the notion of abort chaining, a specific type of attack on fairness of signing protocols, we derive the lower bound α 2 + 1, with α > 2 being the number of signers involved. We obtain the lower bound by relating the problem of developing fair signing protocols to the open combinatorial problem of finding shortest permutation sequences. This relation also indicates a way to construct signing protocols which are shorter than state-of-the-art protocols. We illustrate our approach by presenting the shortest three-party fair contract signing protocol.