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Bob makes a HTTPS GET request to view his personal info on a dating site via a RESTFul web service. He supplies his api key,secret,timestamp and signature to prove that the request came from him. All the header data are encrypted so Bob is very sure no one can eavesdrop on this request.

Alice is a network administrator. She immediately noticed an encrypted request via her Wireshark tool and tries to view the content, but she only sees the encrypted content, the source IP and destination IP.

If Alice submits the exact same encrypted request via her computer (a few seconds after Bob), will she be able to see Bob's personal info?

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Alice's replay request will fail on TLS level because TLS is immune against replay attacks (e.g. look here).

And even if it wasn't: Why the hell would Alice be able to decrypt the response from the server? It would be encrypted by the TLS session key. Just replaying something doesn't give you access to the session key...

kaidentity
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At the start of each TLS session, both the server and the client generate a fixed number of random bytes (using a cryptographically secure RNG) and exchange them; they are called "client random" and "server random". One instance where this random data is used is when calculating the keys for symmetric encryption and authentication, see RFC 5246, section 6.3.

So if Alice simply replays the packets she captured, the server will consider this a new TLS session and generate a fresh server random that is almost surely distinct from the one used during Bob's session, and therefore use fresh symmetric encryption and MAC keys. The ciphertext sent by Alice will fail the MAC check and will be ignored by the server.