Your idea does nothing about people who are ineligible to vote but cast votes anyway.
Let's define "ineligible" voters. In order to vote in a national election, you must be a citizen of legal age, of sound mind, and have no felony conviction, right? Please help me out if there are any other restrictions I've forgotten about.
So the question is, how is most vote fraud perpetrated? Is it by ineligible voters, ie, non-citizens, felons, the underage, and the mentally feeble, voting one time in their precint? I would suggest that the answer is no - most vote fraud is perpetrated through one person voting multiple times (in multiple polling places), either through absentee ballots cast in multiple locations or simply bussing otherwise eligible voters from one precinct to the next.
My solution eliminates the most common forms of vote fraud by ensuring that any individual can only vote once. I should also mention that I agree with a national ID card to help eliminate many of the ineligible voters.