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 based on plaintext equality tests. However, the time complexity of the plaintext equality tests is large, although the mix-and-match protocol can securely calculate any logical circuits. The proposed protocols reduce the number of times the plaintext equality tests is used by replacing them with the Boneh-Goh-Nissim encryption, which enables calculation of 2-DNF of encrypted data.