The problem isn't so much with instituations that use SSN's to uniquely identify individuals, the problem is with institutions that treat an SSN as an authentication mechanism. Identity theft can't be solved by keeping the SSN secret, but it can be made much more difficult by making the SSN is irrelevant, so that it becomes just a way to uniquely identify a person and not a security mechanism.
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The problem isn't so much with instituations that use SSN's to uniquely identify individuals, the problem is with institutions that treat an SSN as an authentication mechanism. Identity theft can't be solved by keeping the SSN secret, but it can be made much more difficult by making the SSN is irrelevant, so that it becomes just a way to uniquely identify a person and not a security mechanism.