Ëëëóòò¹ôöööö Ëëëððð¹¹¹¹ Ùùøøóò Ûûøø Îöö¬¬¬ðð ××ööññòòòø Óó Ô ¼ ¹øø Êóóø

A second-price sealed-bid auction is that a bidder who o ers the highest price gets a good in the second highest price. This style of auction solves the problems of both an English auction and a rst-price sealed-bid auction. An electronic rst-price sealed-bid auction cannot directly be applied to a second-price sealed-bid auction which keeps the highest bid secret. We propose the veri able discriminant function of the p0-th root. Our auction scheme satis es public veri ability of auction results, and also does not have a single entity who knows the highest bid value even after an auction. Furthermore the bidding cost of our scheme is lower than that of the previous one.