[betterunix2]

Actually, index calculus attacks can be applied to certain elliptic curves; for example, supersingular curves. This is one of the reasons why we use standardized curve parameters that have been checked for known weaknesses.

There is also a really interesting class of curves for which the index calculus attack is exactly as hard the "direct" ECDLOG attacks (e.g. Pollard's rho). Those are the "pairing friendly" curves and there are a whole bunch of really interesting applications.