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Quoting from the Eltoo paper:

Before signing and broadcasting the funding transaction the funder requires the other endpoint to create an initial settlement transaction that returns the funds back to the funder. The initial settlement transaction spends the funding output, and creates a single output that returns all of the funds to the funder. This settlement transaction is then signed using the settlement key and returned to the funder. The funder verifies that the settlement transaction matches its expectations, i.e., it returns its funds and is signed by the other endpoint. Now the funder can broadcast the funding transaction, and wait for it to be confirmed effectively starting the contract. The funder also signs the initial settlement transaction, making it complete and returns it to the other endpoint.

In the last sentence, the funder signs the initial settlement and sends it to the other endpoint. But why does the other endpoint need it? This settlement transaction returns all funds to the funder, so there is no benefit for the counter-party to ever publish this first settlement transaction. Couldn't this step be skipped?

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That step could be skipped for the initial settlement tx of a single-funded channel, but then you would need to implement different behaviour for 1) all subsequent settlement txs of a single-funded channel, as well as 2) the initial settlement tx of a dual-funded channel.

In both of these scenarios, the counterparty absolutely needs a copy of the latest settlement tx, to be able to have access to their share of the channel balance.

For simplicity sake, the protocol is kept the same for all three cases instead.

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