How does ECDHE_RSA key exchange mechanism work? Using Wireshark, I found these data exchanged with google.com over TLS:


Client Hello

possible cipher suites and possible curve types (eg. secp256r1) sent

Server Hello

cipher suite selected

Certificate

RSA certificate signatures exchanged, etc. (hopefully, not important for me)

Client Key Exchange, Change Cipher Spec, Hello Request

ECDHE pubkey sent to server

New Session Ticket, Change Cipher Spec, Hello Request, Application Data

session ticket received, etc. (hopefully, not important for me)





Ok, and now my question: What I learned, I need curve equation, random point on it and the finite field size (for modulo operation) in order to apply ECC.

The only thing I found was the pubkey of size 520 bits, which should represent:


1st 8 bits - base 16 prefix
next 256 bits - X coordinate of the "random point"
last 256 bits - Y coordinate of the "random point"


Ok, I have the point, but what about the curve equation and the field size?

Does the curve secp256r1 define only one specific curve? (I suppose not, so where/how is it defined?)

What is and where to find the size of the field? (Or is there only one size for given standard? ...don't think so.)



Thank you for Your answer.

PS: This is how I understood the topic, if I'm wrong, please, correct me. :)
Curve secp256r1 is not a type of curve; it is a curve, and is standardized under that name by SECG, under the name P-256 by NIST, and under the name prime256v1 by ANSI. It also happens to be the by far the most common elliptic curve used in cryptography. The field size, curve equation, and generator point are all part of the curve spec; the point of having a standardized curve is that it takes time to generate elliptic curve parameters, and there's no need for different people to use different curves.

Incidentally, based on your not mentioning the generator point you might be a bit mistaken on how the "random point" in ECDH is created. An ECDH key doesn't just have a randomly selected point for a public key; rather, you randomly pick an integer $k$ as your private key, and then compute the public key as $kG$ for a known point $G$. Like the curve parameters, both parties must know $G$; like the curve parameters, there's no reason why everyone in the world can't use the same $G$. So, this generator point is also part of the curve spec.

You also missed a crucial step in ECDHE: the ServerKeyExchange message. This is a mandatory part of ECDHE exchanges, and comes after the Certificate message and before ServerHelloDone. This has the server's ephemeral ECDH public key; the ClientKeyExchange has the client's ephemeral key.