It's the CMP-MPC protocol from Fireblocks, so there is no private key for the members key, and it has never existed. But n members hold some shares, they can sign the message with their share and combine the signature to form the final signature.
I see. And they do all this through Mixin Messenger, right? Can they do it through any other piece of software, or it has to be your software?
My understanding is that the whole network is dependent on your XIN altcoin and its nodes, of which there are only 20-30? What happens when your network goes down? Does Mixin Messenger go offline? How does the
average user (i.e. one who cannot clone github repos or compile software themselves) manage to recreate the members key and access their coins?
Technically they can use any software to do the CMP-MPC, and even if all the nodes go down, it's technically possible to do the process to access the members key. And all these process doesn't rely on any altcoins, and members should have no knowledge about any altcoins.
Mixin Messenger may go offline and that has never been the case since 2017, and it serves hundreds of millions of E2EE messages everyday.
Our service is to make all this technically possible process easier for average user, when you are trying to think about the backup plan when our service is gone, the plan is not to let the users develop software, it's to provide another software to help them. A decentralized system allows a new software to do the job, unlike a centralized system rug.
An average user can't even use Bitcoin Core to manage multisig and timelock, that's why we provide the service and sell it at a price.
In summary, everything could go offline, and in past 6 years, Mixin Messenger never did that. The software doesn't rely on altcoins. The coins can always be recovered even if our system is completely offline.