  An effective atomic cross-chain swap protocol is introduced by Herlihy
[Herlihy, 2018] as a distributed coordination protocol in order to exchange
assets across multiple blockchains among multiple parties. An atomic
cross-chain swap protocol guarantees; (1) if all parties conform to the
protocol, then all assets are exchanged among parties, (2) even if some parties
or coalitions of parties deviate from the protocol, no party conforming to the
protocol suffers a loss, and (3) no coalition has an incentive to deviate from
the protocol. Herlihy [Herlihy, 2018] invented this protocol by using hashed
timelock contracts. A cross-chain swap is modeled as a directed graph D =
(V,A). Vertex set V denotes a set of parties and arc set A denotes a set of
proposed asset transfers. Herlihy's protocol uses the graph topology and
signature information to set appropriate hashed timelock contracts. The space
complexity of the protocol (i.e., the total number of bits written in the
blockchains in a swap) is O(|A|^2). The local time complexity of the protocol
(i.e., the maximum execution time of a contract in a swap to transfer the
corresponding asset) is O(|V||L|), where L is a feedback vertex set computed by
the protocol. We propose a new atomic cross-chain swap protocol which uses only
signature information and improves the space complexity to O(|A||V|) and the
local time complexity to O(|V|).
