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See http://en.wikipedia.org/wiki/Punchscan for an example of one such system which preserves the secret ballot property.

From reading the wiki article it seems that Punchscan allows the voter to prove which way they voted.

That is incorrect. Why do you think so? Are you thinking of this: ". The voter can look up her ballot by typing in the serial number and she can check that information held by the election authority matches her ballot. "

The voter must retain her vote (A or B) in human memory, which cannot be externally verified, except by brain scan, etc... but that detail is rather unsolvable.

Punchscan is a bit impractical for verifying large ballots, but large contested ballots are rare.

You're right, only the piece you've already kept, which by itself is not enough to prove which way you voted is kept. I was confused because it said that it showed you something which could prove that a vote was 'counted as cast', and I don't see how you can prove that without the system proving to you that it knew how you cast your vote.

Wow! That is a really elegant solution.

I wonder what (if any) pitfalls it has, other than the increased complexity and less obvious correctness (i.e. it would be hard to convince a non-mathematically inclined person that it works properly).

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