To prove my identity without compromising my personal information?
I suppose I could generate a public-private key pair, and, then, showing identification, register my public key with the government. Then I could prove who I am by signing things with my private key.
The thing is that computers get hacked and information stolen. So let's issue everyone an electronic ID card with a little computer in it that is capable of generating a public-private key pair.
There you go; the technical problem is solved. However, generating a public-private key pair with long enough keys to resist being cracked takes a big computer, and if you're giving ID cards with chips in them out free to everyone that won't be used for anything else, you can only really afford to put little computers in them. So now one has to figure out a shortcut.
Well, here's one way to do it:
Take a "big" computer, the size of a desktop PC, or thereabouts, that's powerful enough to generate a secure public-private key pair in a reasonable amount of time like one minute. Set it up so that it's not connected to the Internet. All it has is a connection to the mains, and a slot to plug one's chip ID card into.
And its software is robustly protected against someone putting in a chip ID card with a different chip in it that tries to do a buffer overflow or something to take over the key-generating computer.
So when people take their paper ID down to register their public key to the government office, they first stick their fancy new chip ID card into this computer to get their random public and private key pair. If the government is honest, and hasn't rigged the computer to give them everybody's private key to forge thier signatures, this might even work.