Publicly verifiable auctions with minimal leakage

Publicly verifiable sealed electronic auctions are proposed. The schemes enjoy the following advantages. They require no special trusted parties. After bid opening phase, only the winning price is revealed and the relation of other bidding amounts keep unknown. The winning price is publicly verifiable. Our scheme supports the first-price auction, Vickrey auction and (M+l)st-price auction. Finally, the first-price auction is very efficient and the generic (M+l)st-price auction is applicable when a high standard of security is required.