The desired outcome is to prove that each vote came from a valid voter, while being impossible to prove which vote came from whom.
The desired outcome is to prove that each vote came from a valid voter, while being impossible to prove which vote came from whom.
The coercing party can insist that you take a mobile phone picture of your ballot. This kind of coercion is common in some regions of Russia where civil servants and teachers are told by their superiors that they must vote for the preferred party “or else”, and a mobile phone picture is demanded as proof.
Edit: Perhaps I wasn't clear. Each of the N ballots is already filled out for a different candidate. The envelopes are all identical and sealed by you inside the voting booth. That way an existence proof of a ballot for the given candidate leaks absolutely no information. There are no empty paper ballots, and none of the paper ballots can leave the polling station. Ballots need to be counted both from the "vote" and from the "discard" boxes. Ballots missing from the discard box prove vote tampering.
You go into the voting booth, seal the N ballots in the N identical envelopes, being sure to keep track of where the ballot for your desired candidate is. You leave the voting booth and drop the one envelope in the vote box, and the N-1 envelopes in the discard box.
Even if you're forced to videotape inside the voting booth of your sealing the N ballots in the N identical envelopes, you could pull a quick 3-card-monty when leaving the voting booth. Even a quick random shuffle would be sufficient to statistically nullify your vote, or if you're only able to keep track of where the Dr. Evil ballot is, a quick deal from the bottom of the deck would guarantee a vote for a random opposition candidate. As long as there's some spot between the polling both and the ballot boxes that isn't covered by video, there's no way for a third party to know which ballot went where.
You aren't allowed to leave the polling station with any of the envelopes, and a tally of the ballots in the discard box will show what percentage of people have left the polling station with their discarded ballots. Any non-negligible percentage of "discard" ballots leaving the polling station will make the whole vote suspect. (The discard ballot count will also provide circumstantial evidence for which way the tampering is going, although Dr. Evil will always claim it's false flag vote tampering to make him look bad).
If someone is really coerced in sneaking their discard ballots out of the polling station, they could sneak N-1 ballots out and either put the ballot for Dr. Evil in the discard box or burn it right outside the polling station. They then show the jackbooted thugs the other N-1 ballots. They end up not voting, but at least Dr. Evil is denied the vote.
This requires the poll workers to be somewhat impartial. If they are not all bets are off. In the German Democratic Republic voters where socially pressured (by official policy) to not use the booths at all.
(An exception is when you are a government delegate voting for chancellor, in which case you pay a fine of 1000€, especially after the president has clarified this rule to new delegates repeatedly)
The election council and a lot of privacy-conscious citizens weren't amused to say the least, but a subsequent ruling by a judge turned out that it isn't strictly speaking illegal, because the law doesn't explicitly prohibit photography in the ballot booth. So until parliament enacts a law that forbids it (like Germany), we are stuck with this rather dubious phenomenon.
At least we've banned voting computers for a good while — the ones we had ran closed source software and the votes cast couldn't be verified by the voters or the election council — so there's that.