There is actually a way to authenticate a user over an insecure connection: Secure Remote Password (SRP) protocol. SRP is specifically designed to allow a client to authenticate itself to a server without a Man-in-the-Middle attacker being able to capture the client's password, or even replay the authentication to later impersonate the client, whether the authentication succeeded or not. Furthermore, successful authentication with SRP creates a shared secret key that both the client and the server know, but a MitM does not. This secret key could be used for symmetric encryption and/or HMACs.
However, there are a number of limitations of SRP:
- The user must have registered in some secure fashion, as the registration process does require transmitting password-equivalent material (though the server does not need to store it).
- Although it is safe to attempt an SRP login with an untrusted server (that is, it won't expose your password to that server, and you'll be able to tell that the server didn't have your password in its database), It's not safe to load a login web page from an untrusted server (it could send down a page that captures your every keystroke and sends it off somewhere).
- Although successful authentication via SRP generates a secure shared secret key, the code to actually use this key in a web app would need to be loaded from a server, and an attacker could tamper with that code to steal the symmetric key and make changes to the requests and responses.
In other words, while SRP can be useful in situations where you have a trusted client that doesn't need to download its code over the insecure connection and also you have some other secure way to register the user(s), SRP is not suitable for a web application. Web apps always download their code from the server, so if the connection to the server isn't secure, a MitM (or other network attacker, for example somebody spoofing DNS) can inject malicious code into the web app and there's nothing that you, the victim, can do about it. Once that code is there, it can capture any password or other data you enter, steal any key that is generated, and tamper with any requests you send or responses you receive without you even knowing.