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Elliptic curve cryptography support

I'm trying to use wireshark to debug a problem between NGINX and C#. The problem only manifests itself using EC cryptography, so I can't use RSA keys list mechanism. I tried the (pre)-master-secret log on a connection between my browser and NGINX and managed to decode the connection that way, but I haven't found C# to generate such log.

I've looked around and it seems to be impossible to directly use the private key. If that's the case, I'm curious on why.