There are three participants in our cross-chain virtual payment channel. We name them Alice, Ingrid, and Bob for the easy description. Alice and Bob have coins/accounts only in blockchain systems $B_1$ and $B_2$, respectively. At the same time, Ingrid shares a payment channel with Alice and Bob in blockchain systems $B_1$ and $B_2$, respectively. Our cross-chain virtual payment channel can allow Alice and Bob to perform limitless transactions off-chain. Unlike payment channels, we have the following four kinds of operations in cross-chain virtual payment channels, namely Open, Update, Close, and Offload.

• Open: Suppose Ingrid has payment channels $\alpha$ and $\beta$ in two blockchain systems with Alice and Bob, respectively. Alice and Bob create the cross-chain virtual payment channel $\gamma$ based on $\alpha$ and $\beta$ with the help of Ingrid.
• Update: $\gamma.Alice$ and $\gamma.Bob$ perform instant off-chain transactions by updating the state of $\gamma$.
• Close: $\gamma.Alice$, $\gamma.Ingrid$, and $\gamma.Bob$ reach a consensus on the final state of $\gamma$. With this final state, $\gamma.Ingrid$ updates payment channels $\alpha$ and $\beta$ with $\gamma.Alice$ and $\gamma.Bob$, respectively. After that, the cross-chain virtual payment channel $\gamma$ is closed.
• Offload: $\gamma.Alice$, $\gamma.Ingrid$, and $\gamma.Bob$ execute the offload procedure when some dispute happens in other operations of the channel $\gamma$. $\gamma.Ingrid$ closes payment channels $\alpha$ and $\beta$ with $\gamma.Alice$ and $\gamma.Bob$, respectively. Furthermore, the cross-chain virtual payment channel $\gamma$ will be closed and become a payment channel in one of the underlying blockchain systems. In the end, the dishonest participant of $\gamma$ will be punished.

Security Goals

Since the cross-chain virtual payment channel is essentially a payment channel that has not been published on the blockchain, it must have the security properties of the payment channel, including

• Consensus-Open. A channel can only be successfully opened when Alice, Bob, and Ingrid reach a consensus on the open of the channel.
• Consensus-Update. The state update of the channel needs to be approved by Alice and Bob.
• Consensus-Close. A channel can only be successfully closed when honest users agree on the consequence of the closure, and dishonest users are penalized.

Efficiency goals

As the existing virtual payment channel schemes, cross-chain virtual payment channels should also satisfy the following efficiency goals in terms of communication rounds.

• Efficiency-Open. Constant round opening: Successful open of a cross-chain virtual payment channel $\gamma$ takes a constant number of rounds.
• Efficiency-Update. Optimistic updating: For the channel $\gamma$, this property guarantees that a channel update takes a constant number of rounds if $\gamma.Alice$ and $\gamma.Bob$ are honest.
• Efficiency-Close. Optimistic closing: When all participants in the channel $\gamma$ are honest, the closing of $\gamma$ takes a constant number of rounds.

Property Efficiency-Update applies to all kinds of off-chain channels, including payment channels, virtual payment channels, and our proposed cross-chain virtual payment channels. Furthermore, in the optimistic case, all the operations in (cross-chain) virtual payment channel are executed off-chain.