I've heard that a modified version of Shor's algorithm can "break" ECC. But what does this mean specifically? What are all the things you can do with this algorithm? Can you:
- decrypt messages encrypted with the private key? Does the original message have to be in a certain recognizable format in order to do this?
- encrypt messages to look like they were encrypted with the correct private key (without actually knowing the private key)?
- forge signatures?
- Discover the actual private key via just an encrypted message?
- Discover the private key via just a signature and the message it signs? Just the signature?
And are there other ways of "breaking" ECC that have additional properties?