Authenticated key agreements enable users to determine session keys, and to securely communicate with others over an insecure channel via the session keys. This study investigates the lower bounds on communications for three-party authenticated key agreements and considers whether or not the sub-keys for generating a session key can be revealed in the channel. Since two clients do not share any common secret key, they require the help of the server to authenticate their identities and exchange confidential and authenticated information over insecure networks. However, if the session key security is based on asymmetric cryptosystems, then revealing the sub-keys cannot compromise the session key. The clients can directly exchange the sub-keys and reduce the transmissions. In addition, authenticated key agreements were developed by using the derived results of the lower bounds on communications. Compared with related approaches, the proposed protocols had fewer transmissions and realized the lower bounds on communications.