Yes! SSNs are already not private given the number of hacks that have occured. Today, the real damage comes from the fact that people/businesses still believe they are private. Publish a list of all SSNs would eliminate the misperception once and for all and force people to verify identity in a better way. SSNs should only ever be used for your employer knows how to report who paid what taxes to the IRS. If someone else wants to use my SSN to claim that they paid my taxes, fine with me!