To clarify the Windows crypto fail: The problem isn't in signature validation. The problem is the *root store/cache*. CryptoAPI considers an (attacker-supplied) root CA to be in the trust store if its public key and serial match a cert in the root store, Ignoring curve params.
Is it easier to find params that generate an arbitrary pub key than to find a pub key given the params? Aren't both roughly just at hard? What am I missing?
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What you're trying to find is the private key given the public key. You cannot find the original private key for the original params, but you can trivially craft parameters in such a way to make a private key of 1 "happen" to correspond to the original public key.
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