[dc352]

I suspect you're overcomplicating the attack with all the math and we can ignore most of it.

The only way the attacker can tell the MS Crypto API is via the TLS protocol. You can only do it if it's relevant. The only option for that is to use ECDH, which allows the server to supply EC parameters for the Diffie-Hellmann exchange.

My bet is that the problem is that MS Crypto API took those parameters as correct without checking them against what's in the certificate. I.e.,

ServerKeyExchange - here's the EC spec, we just need the public key Certificate - ah - here's public key, we have the ECparams - let's run the math

:)