Alice and Bob want to exchange keys, Alice sends his half (the public key of the ephemeral key pair) signed with his private key and makes the public key available. Eve is bridging the connection so he intercepts the message uses the public key and verifies the half. He then sends his half to Alice signed and establishes a connection. Now Alice is communicating with Eve and Eve is decrypting the message and sending to Bob.
Isnt this exactly same as the raw Diffie Hellman ? How are we exactly preventing man in the middle attack if Alice thinks the whole time that Eve is Bob ? Is it correct to assume that the message(half key) is not encrypted ? Is a certification authority also required for ephemeral diffie hellman ?
Please correct me if I have misunderstood Ephemeral Diffie Hellman