This paper presents efficient secure auction protocols for first price auction and second price auction. Previous auction protocols are based on a generally secure multi-party protocol called mix-and-match protocol. However, the time complexity of the mix-and-match protocol is large, though it can securely calculate any logical circuits. Our auction protocols reduce the number of times the mix-and-match protocol is used by replacing them with the Boneh-Goh-Nissim encryption that enables calculation of 2-DNF of encrypted data.