I was asked to do a research about attacks on various digital signature schemes. There is one thing that bothers me for which I could not find a direct answer.
For digital signatures that are based on prime numbers (like RSA) attacks are theoretic for example "chosen ciphertext attack" or "known plaintext attack", a very good source could be "Twenty years of attacks on RSA cryptosystem"
For Elliptic Curve Cryptography (ECC) based digital signatures attacks are purely physical, for example "fault attacks" or "side-channel attacks", a typical source is "A survey of fault attacks"
Why there are no theoretic attacks on ECC? the reason that answer is important for me is because I am supposed to implement an attack for a specific ECC digital signature. Since it is not implemented on an actual device (at least yet) I can't do that (attacks are physical) and I can't figure another similar work as my guideline.
I barely managed to find some few paragraphs like this "The main reason for the attractiveness of ECC is the fact that there is no sub-exponential algorithm known to solve the discrete logarithm problem on a properly chosen elliptic curve" is this the reason? or am I missing something very obvious?