The base concept I was thinking of would be that usernames on this system would be numeric, a bit like ICQ numbers. People apparently didn't mind being assigned some "random" number that they logged in with and gave out to friends. But, unlike ICQ numbers, the numeric IDs on this system would actually serve a purpose (instead of being simply the auto-incrementing primary key in ICQ's database).
Each user's client would generate an RSA public/private key pair, which is obviously the foundation for any sort of secure messaging platform. Then, you'd take the fingerprint of the public key (a short hexadecimal string, which could look for example like "45:2f:a5:d8:13:95:ba:03:51:c4:8d:ac:82:a8:4c:6a"), and you'd turn that into an integer number. Then you'd take, say, the first and last 5 digits of the number to create a 10 digit "screen name" number.
For the sake of continuing my description, let's pretend that the number came out to 567-8426-789. When setting up the chat client for the first time, it generates keys behind the scenes, and says "Your login number is 567-8426-789. You can give this out to your friends so they can add you to their buddy lists."
This idea stems from the general practice where, suppose you downloaded a Linux OS DVD, and you wanna verify the download so you look up the
sha256sum for it. You're not going to compare the two SHA-256 hashes character by character; you'll check that the first 5 or 6 digits are the same, and the last 5 or 6 too. If those are, you can reasonably assert that your copy of the DVD is the same as the intended one. The numeric login ID of this secure IM network would be similar.
By giving out your ID number to a friend, they're automatically verifying your identity, since your ID is based off your public key. If somebody was trying to impersonate you (with a completely different RSA key pair), they wouldn't likely end up with the same numeric ID so it wouldn't work. ;)
Of course, the IM clients themselves would include features to do a more thorough verification of the friend you just added, like allowing you to see their full hexadecimal fingerprint, and also show a visual fingerprint to make it even easier to verify at a glance that your friend is who you expect them to be.
Anyway, this is just the basic idea I had. As far as syncing other devices to use the same account goes, it would probably need something like the old/current method Firefox Sync uses, i.e. needing to enter a "random" code from one device into the other so that your private key can be securely transferred between them without a third party in the middle being able to see. Alternately, the system would be federated (so you could host your own servers, like XMPP), and your local server may be trusted with an encrypted copy of your private key, encrypted using your log-in password. But these are all details to be figured out later.
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