# Atomic Cross-Chain Swaps with Improved Space and Local Time Complexity

## TL;DR

This paper introduces an improved atomic cross-chain swap protocol that reduces space and local time complexity by using only signature information, enhancing efficiency in multi-party blockchain asset exchanges.

## Contribution

The paper presents a novel protocol that decreases space complexity to O(|A||V|) and local time complexity to O(|V|) compared to Herlihy's original method, using only signature information.

## Key findings

- Reduces space complexity from O(|A|^2) to O(|A||V|).
- Reduces local time complexity from O(|V||L|) to O(|V|).
- Maintains protocol guarantees despite complexity improvements.

## Abstract

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|).