[kwantam]

Maybe we're looking at different things, but the link appears to discuss ElGamal encryption, which is discrete log based (which means modern implementations use elliptic curves; historically it would have been discrete log in a subgroup of a large prime field). It also talks about BLS signatures, which are exclusively elliptic curve based.

By and large, anything whose security relies on discrete log can be implemented using an elliptic curve, but beginning cryptography classes treat that as an implementation detail because mostly all you need is a prime-order group, and elliptic curves can mostly be treated as a black-box prime order group.

(BLS signatures are an exception; they require a bilinear pairing, which in turn requires a special kind of elliptic curve that's not just a black-box prime order group.)

There are all sorts of great algebraic geometry tricks to be played with elliptic curves, but those almost certainly aren't going to be found in an intro crypto class, or maybe any CS class...