There's a lot of things that use public/private keys however, or security tokens, or whatnot. Should it be a smartphone app? A hardware dongle? Etc. etc. If a hardware dongle, which one?
As such, its the Executive Branch's job to research the various technologies, and implement a new standard to solve the online identity problem.
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For example, 18f (White House's crack website team) has the following: https://login.gov/
Github code here: https://github.com/18F/identity-idp
If single-sign on were widely deployed across US Agencies (and tied to financial services / private sector banks), we'd be in a way better place.
In any case, this is clearly the realm of the Executive Branch. Specifically 18f probably should continue to lead the effort, as they have been.
SSNs are fine and useful. They just shouldn't be the "password" to financial systems. When every damn bank uses "Whats your pin and SSN" to gain access to an account... that's the problem.
The issue is that private companies use SSNs for security. There's nothing wrong in using SSNs as an identifier.
The USA needs to start assuming that SSNs are public information, and to build security through other means. SSNs were never a secret number to be used for authentication / authorization purposes.