Even if something is open source, if it's running on a server you don't control there's no guarantee that the service you're talking to is running that same code as what you've read.
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The whole point of the app is end-to-end encryption instead of trusting the server with the data.
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That's the idea, but the problem is verification--there's a black box involved in the process. How do you know the mechanism inside the box produces end-to-end encryption and not merely apparently end-to-end encryption?
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There is no black box involved in the process. The Signal app is open source. The whole point of the app is providing end-to-end encryption from client to client. You seem to be confusing end-to-end encryption with transport encryption. It doesn't mean what you seem to think.
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It uses authenticated encryption with forward secrecy between instances of the app. It doesn't trust the server. Encrypting connections to the server is not end-to-end encryption. End-to-end means encrypting from one end (Signal app) to the other (Signal app), not to the server.
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The app sets up an encrypted session with the other end. It provides forward secrecy rather than using long-lived keys directly for encryption. You either verify the safety number with contacts out-of-band to verify a MITM didn't happen for the initial setup or rely on TOFU.
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If you verify the safety number, it starts considering it a secure session and marks it as secure. It will prominently notify you if the encryption has to be renegotiated.
If you don't ever verify a session, it still informs you if it renegotiates (safety number changed).
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So, for people who don't care, they still get TOFU and a notice is given if it has to start over. For people willing to put in the minimal effort of scanning the safety number QR code out-of-band, it enforces it.
If you use the encrypted backup/restore, you can move it across phones without losing your data including not requiring new safety numbers. A lot of people don't transfer over their data with the app's backup feature when they switch phones, so their safety numbers change.