I think we need to distinguish between the private key associated with the certificate, and the session key. With RSA based private keys there is no elliptic curve involved at all (so none needs to be chosen or included in the certificate). However, RSA private keys (used for authentication) can be used with EC for encryption - both ECDH and ECDHE. Both are DH based session key establishment methods, and both use elliptic curves. And in both cases an elliptic curve needs to be either negotiated or pre-agreed. Which curve is used with ECDH and RSA keys/certs? You claim the curve with ECDH is derived somehow of the RSA public key? Mmmh. Also: why is there then a set of elliptic curves _announced_ by clients at all?