The hardness of discrete logs over fields doesn't have much to do with the hardness of elliptic curve discrete logs. At this stage, I don't think we have any ev

The hardness of discrete logs over fields doesn't have much to do with the hardness of elliptic curve discrete logs. At this stage, I don't think we have any evidence that curves over binary fields are less secure than over prime fields, especially for cryptographically relevant curve sizes. (The situation is different for pairing-friendly elliptic curves, of course, but that's a different kettle of fish).