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I am reading information related to SSL handshake on the Internet. It seems like in the last step the domain name is compared to the domain in the certificate.

How exactly is this protecting against a malicious DNS routing? If the user is diverted to a malicious IP, the domain will still match. I was expecting a challenge response using the public key of the server, but I cannot find it. Are we just relying on whether the response data from the server decrypted with the proposed symmetric key is gibberish/incomprehensible?

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How exactly is this protecting against a malicious DNS routing?

Not at all.

If an evil guy has a valid certificate (e.g. from a hacking a CA) and then manages to man-in-the-middle you, then your connection is hacked.

I was expecting a challenge response using the public key of the server, but I cannot find it.

In order to do this you would have to know some things about the server first. And you don't usually have that info. But certificate pinning is one way of communicating that info:

Certificate pinning helps

Certificate pinning allows you to give a client some extra info about your certificates. Such as: "For my certs, please only accepts signatures by ACME-CA." Or "For my certs, please only accept them if they contain the following public-keys ...".

Further reading for pinning: Wikipedia: HTTP Public Key Pinning

EDIT:How are key exchanges authenticated?

From the comments section:

I am exactly asking how the server proves that he has the private part. It should be a challenge/response initiated by the client using the public key of the server (or using the symmetric key once it is established). I cannot find this in the texts

Indeed, there is challenge/response here.

This is detailed in a subsection of the TLS 1.2 RFC.

There are three options:

  • F.1.1.1. Anonymous Key Exchange
    No authentication at all.
  • F.1.1.2. RSA Key Exchange and Authentication
    The client encrypts (part of) the bulk encryption session key with the server's public key. The server's ability to decrypt is proof of possession of the private key.
    No "Forward secrecy":
    If the server's private key is EVER exposed, even years late, then all old session keys can be retroactively decrypted.
  • F.1.1.3. Diffie-Hellman Key Exchange with Authentication
    Client and server use a different key agreement protocol. The server then signs the a transcript of the negotiations with its private key. If the client can verify this signature with the server's public key, then that is proof that the server really knows the private key.
    Has "Forward Secrecy":
    The advantage of this procedure is "Forward secrecy". If the server's private key is ever exposed years later, then session keys can NOT be retroactively decrypted. (Because, well, they were never encrypted with the server's private key in the first place. See Diffie–Hellman key exchange for details.)
  • I am not considering the scenario of hacking a CA. You can get the certificate directly from the real web server. You just ask for it and it provides it to you. You take it and put it to another web server and reroute the DNS. – Cowboy Trader Jul 2 '15 at 11:38
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    Yes. But the certificate is only one half of the certificate/private-key pair. And the server keeps the private part. But it can prove to you that it knows the private part without actually revealing it to you. See this post for details: How does SSL/TLS work? – StackzOfZtuff Jul 2 '15 at 11:43
  • I am exactly asking how the server proves that he has the private part. It should be a challenge/response initiated by the client using the public key of the server (or using the symmetric key once it is established). I cannot find this in the texts – Cowboy Trader Jul 2 '15 at 11:55
  • I see. Yes, there is challenge/response there. See added section. – StackzOfZtuff Jul 2 '15 at 12:32

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