ConsenSys has published a paper that details a method of connecting public and private blockchains through a new Atomic Crosschain Transaction technology. They say:
“We propose a crosschain technique which allows transactions to be executed atomically across sidechains, introduce a new mechanism for proving values across sidechains, describe a transaction locking mechanism which works in the context of blockchain to enable atomic transactions, and a methodology for providing a global time-out across sidechains.”
The paper goes into step by step instructions on how to establish this interoperability system, but at a high level conceptual view, we can first consider the Lightning Network (LN) and how it is kind of a sidechain.
LN uses Hashed Timelock Contracts. That is you lock your bitcoin in a multi-sig address (channel) and get a hash that proves how much you have locked. You can share this hash with the receiver who then agrees to the transfer.
The LN nodes, plus bitcoin nodes, can see there was an exchange and an agreement, thus process the transaction and enforce the contract.
Ethereum has smart contracts. So here you have a smart contract on Blockchain A and a smart contract on Blockchain B. You lock the eth on one blockchain, you present the proof of how much was locked, and you unlock it on the other blockchain.
Here too you need to run the node of both Blockchain A and B. In this paper they call them MultiNodes, so you can have C, D… whichever chain you are interested in, transacting in, or validating.
To connect these two smart contracts, there’s a cross-chain transaction. This will get a bit technical, but the paper says:
“A Crosschain Transaction consists of an Originating Transaction and one or more Subordinate Transactions and Subordinate Views, where the Originating Transaction is the Ethereum Transaction which executes on the sidechain on which the Crosschain Transaction was submitted, and the Subordinate Transactions and Subordinate Views are Ethereum Transactions and Ethereum Views which execute on other sidechains as a result of the Originating Transaction.”