There are theoretical cryptographic systems where each voter can verify that his vote was counted properly, without revealing his vote to anyone. I don't think any have been implemented in practice.
That in itself is a problem because the ability for the voter to prove who they voted for opens them to coercion or bribery. Although doing it online (or by post) opens that risk anyway.
I read a paper on one that allowed up to a randomly selected 50% of votes to be audited and still preserve the secret ballot, but it was so complicated that I barely followed and I definitely don't think I could convince a room full of people it was safe.
This is a critical distinction. As a concrete example, here's how voting worked in a scheme I once read about. On any one ballot, the order of candidates was randomized. Then the way the scheme worked was that after voting, the voter tore off the candidate positions (but not their vote) and threw it away in a huge pile of them, burned it, or whatever. (Made it so that someone couldn't come behind them and figure out their position list, essentially.)
Later, after the votes were tallied, the voter could verify that their ballot was (1) counted and (2) counted towards their chosen candidate. But crucially, all they could verify was that the vote counted towards position 1, or position 2, or position 3, ...
The point is that since the voter couldn't prove to a coercing party that the position they voted for was (or was not) the candidate the coercer wanted them to vote for, they were immune to coercion. They could prove that they voted for position 2, sure. But which candidate was at position 2?
The voter knows the truth because they saw the position list. However, until we have mind-reading technology, a coercing party could only take the voter's word.
I'm not following how the counting is done. If all the counter has is a ballot with position 2 checked and the corresponding candidate name torn off, how does that vote get tallied to the proper candidate?
That doesn't verify anything. They are shuffling encrypted data between devices, but none of that is connected to the actual results. This isn't verification, it's smoke and mirrors.
That's not the point of this feature. The point is that if your computer is infected with malicious software that blocks or manipulates your votes then you can detect such things.
The International Association for Cryptologic Research (IACR) uses Helios Voting [1], an implementation of a cryptographic voting protocol [2], to vote for its directors. See the 2010 mock election [3,4] or the 2012 vote for the IACR directors [5]. You can find some other technology resources for Helios here [6].
