Multi-party fair exchange protocol

In cryptography, a multi-party fair exchange protocol is protocol where parties accept to deliver an item if and only if they receive an item in return.¹

Matthew K. Franklin and Gene Tsudik suggested in 1998² the following classification:

• An ${\displaystyle n}$-party single-unit general exchange is a permutation ${\displaystyle \sigma }$ on ${\displaystyle \{1...n\}}$, where each party ${\displaystyle P_{i}}$ offers a single unit of commodity ${\displaystyle K_{i}}$ to ${\displaystyle P_{\sigma (i)}}$, and receives a single unit of commodity ${\displaystyle K_{\sigma ^{-1}(i)}}$ from ${\displaystyle P_{\sigma ^{-1}(i)}}$.
• An ${\displaystyle n}$-party multi-unit general exchange is a matrix of baskets, where the entry ${\displaystyle B_{ij}}$ in row ${\displaystyle i}$ and column ${\displaystyle j}$ is the basket of goods given by ${\displaystyle P_{i}}$ to ${\displaystyle P_{j}}$.