An Algebraic-Topological Approach to Processing Cross-Blockchain Transactions

Abstract

The state-of-the-art techniques for processing cross-blockchain transactions take a simple centralized approach: when the assets on blockchain $X$, say $X$-coins, are exchanged with the assets on blockchain $Y$---the $Y$-coins, those $X$-coins need to be exchanged to a "middle" medium (such as Bitcoin) that is then exchanged to $Y$-coins. If there are more than two parties involved in a single global transaction, the global transaction is split into multiple local two-party transactions, each of which follows the above central-exchange protocol. Unfortunately, the atomicity of the global transaction is violated with the central-exchange approach: those local two-party transactions, once committed, cannot be rolled back if the global transaction decides to abort. In a more general sense, the graph-based model of (two-party) transactions can hardly be extended to an arbitrary number of parties in a cross-blockchain transaction. %from why to how In this paper, we introduce a higher-level abstraction of cross-blockchain transactions. We adopt the \textit{abstract simplicial complex}, an extensively-studied mathematical object in algebraic topology, to represent an arbitrary number of parties involved in the blockchain transactions. Essentially, each party in the global transaction is modeled as a vertex and the global transaction among $n+1$ ($n \in \mathbb{Z}$, $n > 0$) parties compose a $n$-dimensional simplex. While this higher-level abstraction seems plausibly trivial, we will show how this simple extension leads to a new line of modeling methods and protocols for better processing cross-blockchain transactions.