How does performing scalar multiplication on an elliptic curve compare to exponentiation in a multiplicative group modulo a prime?
I.e. on a given elliptic curve of size $|t|$, what's the complexity of computing $kP$ with respect to $|k|$ and $|t|$?
And in comparison, what's the computational complexity of a modular exponentiation of form $g^a \bmod p$ with respect to $|g|=|p|$ and $|a|$?
It would already be a big help to know the answer for double-and-add and square-and-multiply respectively.