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When the key exchnage is DHE, the server key exchange message contains the key paramteres: g^y (the server's public DHE value), g, p. Can you clarify to me what is the server key exchange content if the key exchange is ECDHE? What paramters are included there?

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TL;DR: The public key and the name of the group.

RFC 4492 (especially section 5.4) has all the details for you.

What you basically do is the following:

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