The CBC - MAC mode is not a parallel one. A parallel authentication mode (PKCB) based on double blocks was put forward in this paper. The PKCB mode had a marked improvement on security & speed over parallel authentication mode, PMAC. And it may be combined with the CTR (counter) encryption mode to form a full block cipher mode. On this ground, another parallel authentication mode (KCTR - MAC) based on key counter was advanced. As compared with the PMAC mode, the KCTR - MAC mode had a marked improvement on security, while its speed did not become lower. The KCTR - MAC authentication mode may be combined with the CTR (counter) encryption mode to form a full block cipher mode (2CTR)，too. The 2CTR mode had a performance advantage over the standard mode, CCM (CTR with CBC - MAC). And it was a fast, practicable mode with security.