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Summary form only given. I consider trading performance of a protocol against small, well-defined, security weaknesses. The weaknesses will be of the form that a hacker with some knowledge can, with well-defined probability, guess how a voter voted or influence the election in the way he desires. By setting parameters, these probabilities can be made as small as desired with the cost of additional computational effort. The constructions are quite general, but are presented in the context of the generalized Paillier crypto system with a particular system of noninteractive zero-knowledge proofs for implementing a secure voting protocol.