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From my research, I think what everything says is that the result of a zero knowledge proof (ZKP) is that the participants end up with a shared secret that can be used as a session key for a further symmetric encryption session between those two parties. And then, all of the articles say that you follow this up by starting up something like a Diffie-Helmann (DH) session.

If I just got a secure key from the ZKP - why do I need DH? Why don't I just expand the ZKP result and go from there?

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    $\begingroup$ Whatever those sources are, you shouldn't use them. ​ ​ $\endgroup$ – user991 May 9 '17 at 23:08
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From my research, I think what everything says is that the result of a zero knowledge proof (ZKP) is that the participants end up with a shared secret …
If I just got a secure key from the ZKP - why do I need DH? Why don't I just expand the ZKP result and go from there?

You obviously misunderstood ZKP at its core.

Zero knowledge proofs are interactive or non-interactive protocols or schemes for proving membership of a common input in a formal language (or the analogous thing for promise problems).

In the basic ZKP setting, there are only two parties, in which case something that is unknown to one party ​is simply not a shared secret. For proofs (or for arguments), the verifier is only necessarily convinced that a valid secret exists, not that the prover has one. Also see proofs of knowledge. ​ ​ ​ ​ ​ ​ ​ ​ ​

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