I'll try. There are a lot of ways to flesh out the details of such a system. The essence of all of them, though, is that each person has a public key which is

I'll try. There are a lot of ways to flesh out the details of such a system. The essence of all of them, though, is that each person has a public key which is known to everyone and simultaneously serves as a unique identifier to that person and a private key that is known only to the individual. Additionally, either the public key is derivable from the private key or both keys are derivable simultaneously via some process, but importantly, the private key is not derivable from the public key. And, depending on the system, messages can be encrypted with the private key and only decrypted with the public key, or vice-versa, messages can be encrypted with the public key and decrypted with the private key. The name for this broad category of cryptographic systems that have these properties is called "asymmetric cryptography". To give a specific example of how it could work, each person creates a public-private RSA keypair, and publishes the public key. If I want to apply for a credit line with my bank, my bank can read the publicly available public key and use it to encrypt a nonce (secret token) and send me the encrypted nonce. I decrypt it using my private key and send the original nonce back to my bank, proving that I am the person in control of the private key, without ever revealing what that key is. Thus, if a malicious employee at my bank wanted to impersonate me to some other company, he wouldn't be able to (which is a huge flaw with the current system of handing over secrets to prove identity). To tie public-private keypairs to real humans, you'd go to the DMV and give the government all the proof of identification that is required to receive a driver's license along with your public key, and the government could then publish that you're really the person associated with your public key, and all this still without the government having the knowledge required to impersonate you.