We consider the problem of cross-chain payment whereby customers of different escrows---implemented by a bank or a blockchain smart contract---successfully transfer digital assets without trusting each other. Prior to this work, cross-chain payment problems did not require this success, or any form of progress. We demonstrate that it is possible to solve this problem when assuming synchrony, in the sense that each message is guaranteed to arrive within a known amount of time, but impossible to solve without assuming synchrony. Yet, we solve a weaker variant of this problem, where success is conditional on the patience of the participants, without assuming synchrony, and in the presence of Byzantine failures. We also discuss the relation with the recently defined cross-chain deals.
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