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This paper examines two-pass authenticated key exchange (AKE) protocols that do not use the NAXOS technique and that are secure under the gap Diffie-Hellman assumption in the random oracle model. Their internal structures are also discussed. We introduce an imaginary protocol, however insecure, to analyze the protocols and show the relations between these protocols from the viewpoint of how they overcome the insecurity of the introduced protocol. In addition, this paper provides ways to characterize the AKE protocols and defines two parameters: one consists of the number of static keys, the number of ephemeral keys, and the number of shared values, and the other is defined as the total sum of these numbers. When an AKE protocol is constructed based on some group, these two parameters indicate the number of elements in the group, i.e., they are related to the sizes of the storage and communication data.