Electronic voting is a prime application of cryptographic tools. Many researches are addressing election or confidence voting in this area. We address a new type of voting scheme "Divisible Voting Scheme," in which each voter has multiple ballots where the number of ballots can be different among the voters. This type of voting is popular. We first define the divisible voting scheme and show naive protocols based on existing voting schemes. Then we propose two efficient divisible voting schemes. The first scheme uses multisets, the second scheme uses L-adic representation of the number of ballots. The total cost for a voter is O(M 2 log (N)) in the first scheme and O(M log (N)) in the second scheme where M is the number of candidates to vote for and N is the number of ballots for a voter.

The auction is a popular way of trading. Despite of the popularity of the auction, only a small number of papers have addressed the protocol which realize the double auction. In this paper, we propose a new method of double auction which improves the algorithm of the existing double auction protocol. Our new method is based on the idea of number comparison which is realized by homomorphic encryption. The new method solves the problem of the privacy of losing bids found in the existing algorithm. The buyers and the sellers can embed a random number in their bidding information by the use of the homomorphic encryption. The players in an auction cannot get anyone else's bidding information. The new method is more efficient than the existing ones. Our new method satisfies the criteria for the auction protocol.

This paper provides a M+1-st price auction scheme using homomorphic encryption and the mix and match technique; it offers secrecy of bidding price and public verifiability. Our scheme has low round communication complexity: 1 round from each bidder to auctioneer in bidding and log p rounds from auctioneer to trusted authority in opening when prices are selected from p prefixed choices.