An anonymous password-authenticated key exchange (anonymous PAKE) protocol is designed to provide both password-only authentication and user anonymity against a semi-honest server, who follows the protocol honestly. Very recently, Yang and Zhang [25] have proposed a new anonymous PAKE (NAPAKE) protocol that is claimed efficient compared to the previous constructions. In this paper, we propose a very-efficient anonymous PAKE (called, VEAP) protocol that provides the most efficiency among their kinds in terms of computation and communication costs. The VEAP protocol guarantees semantic security of session keys in the random oracle model under the chosen target CDH problem, and unconditional user anonymity against a semi-honest server. If the pre-computation is allowed, both the user and the server are required to compute only one modular exponentiation, respectively. Surprisingly, this is the same computation cost of the well-known Diffie-Hellman protocol that does not provide authentication at all. In addition, we extend the VEAP protocol in two ways: the first is designed to reduce the communication costs of the VEAP protocol and the second shows that stripping off anonymity parts from the VEAP protocol results in a new PAKE protocol.