Here of course is where our protocol succeeds. Our protocol achieves savings in both computational load and message complexity.
In terms of computational load, each intermediate level authority now handles only 1/D the number of voters as in Protocol 7, leading to a direct factor of Dreduction in computational cost. Some of this is given back by the encryption needed in our modified Protocol 8, and by the computational cost incurred by higher level authorities. Still, these should be small compared with D. The computational cost of voters also decreases, since in Protocol 7 voters are given the tasks of verifying the interactive proofs of the other voters and the final computations of the authorities, both of which scale by the number of voters. The voters then also reduce their computational load by D.
Our protocol also sends only 1/D as many messages as Protocol 7 does. In Protocol 7, each voter needs to broadcast to all other voters, which results in message complexity quadratic in the number of voters. In an election district with 1/D as many voters, only 1/D2 as many messages are sent. Over D election districts, this results in a total savings of 1/D.