I want to implement an internet-based e-voting system. Voters shall be able to cast their vote for one out of n possible candidates. Each candidate has his own ballot-box kept by and at a trustworthy third party. This third party is absolutely trustworthy insofar as it can be trusted to attend to his duties of supervising the cast diligently. However it is not without bias and hence under no circumstances must be able to see which voter voted for which candidate.
This problem can be approached using homomorphic encryption. Votes are homomorphically encrypted and homomorphically added to the ballot-boxes. In order to prevent the third party from gaining any knowledge of any vote cast every voter puts one vote into every ballot-box. n-1 times this vote will be "0" for "no vote for this candidate" and only one time it will be "1" for "vote this candidate".
One unpleasant side effect of this approach is that the third party apparently is not able to check the correctness of every single vote. A voter may try to give a "1" to more than one candidate or a "2" to a single one. Of course the third party could hand out pre-signed voting coins in advance that the voters would use for casting their votes. However the third party would recognize these coins, i.e. their values, and hence still know who voted for whom.
Do you have any idea how to solve this problem relying on as few possible other (trusted) third parties as possible?