It seems that tying your SSN to your vote would make sense. A computer could easily tell if there were multiple hits on one number, (voting more than once, or coming back from the dead). All votes for that SSN would then be voided and in investigation started. The real person identified and if determined a victim of ID theft allowed to recast their ballot (once).
Cross linking SSN with voter names would be VERY effective.
So they won't.
You don't want any system that ties a particular vote to an individual. That would be the death of secret ballot. There is a reason for keeping those things separate.
Mainly, if your vote CAN be traced to you, you can be pressured to vote a certain way. Or, you could reliably sell your vote, since you have the means to guarantee what it was.
Nope, the way to stop the fraud is to stop the fraudulent votes from entering the system in the first place. After they get in, you're stuck with them.
Actually, I think this would be illegal. SS# is not supposed to be used by the government for other purposes.