> I didn't understand the algorithm completely (for example, whether you post your vote anonymously or under your name, and where is the private key for the public key you mentioned).
It's best to illustrate it with the ElGamal cryptosystem. Let's say that system officials have set up keypair `sk`, `pk = g^sk` and let everyone know `g, pk`. To submit a vote, the voter selects an option corresponding to a message `m` and encrypts it with a freely chosen randomization factor `r` and obtains a tuple `(g^r, m*pk^r)`. He signs this encryption under their name and sends it to the bulletin board.
The last bit is whether to allow everyone to see that you have or have not voted so whether the fact that you have participated in the elections. There seems to be the consensus in the literature that the signature should be concealed from the public and be allowed to verify only for independent auditors.
> Let's say bulletin board software replaces the vote with 20% probability. You post your vote for candidate A and see that it didn't appear on the board (because the board replaced it with vote for candidate B, but you don't know about it). How can you prove that you tried to vote for A and not for B? The records show that you have voted, and as voting is anonymous it is impossible to know how you voted.
One way to preserve the integrity of the bulletin board is that upon receiving a `vote <- ((g^r, m*pk^r), sig)`, the bulletin board issues a signature on the `vote` and returns it to the voter for the latter to assert for the vote to not be changed. Even when the signature is not present on the bulletin board, the voter can check the presence of `(g^r, m*pk^r)` as the randomization factor makes it unique for each voter.
> Of course, there are other ways to meddle with such election. For example, you see that the turnout is 99%. How can you verify this number? The government refuses to publish a list of voters because GDPR doesn't allow that. And even if the country publishes this list how you can verify that the list doesn't contain fictious voters?
The fictitious voters are indeed a thing if we can't trust the independent auditors of the bulletin board. Personally, I would never support an internet voting system where the result of the elections would lay on the integrity of a few trusted auditors who have special access to do so. Thus I would greatly prefer for the voter lists (the signatures) to be public in spite of losing participation anonymity.