I'm definitely opposed to 'electronic' voting, but the problem you are describing is easily solvable with asymmetric cryptography.

You'd have your private key on your 'voting card' (unsure how you call it in the US) with some pw and the machine would just scan it, ask your password. You just need the proof that someone having a private key associated with a public key on a public record has signed, not necessarily know to which public key it is associated.

But there would be so much problems with such a system that I would'nt even know where to start..