[openasocket]

For your clarification, Shor's algorithm just involves encoding every possible value of a function (say f(x) for all values of x) into a bunch of qubits, taking the fourier transform, and measuring, which will tell you the period of that function (some r such that f(x + r) = f(x) for all x). Thus, you can find the period of any function that you can define a fourier transform for. For breaking RSA, your function is on the domain of integers modulo n, so you use the discrete fourier transform. You can also define a fourier transform for functions on the domain of an elliptic curve as well! In fact, any finite abelian group has a unique definition of fourier transform (see https://en.wikipedia.org/wiki/Pontryagin_duality for some of the math behind it). That means we can't just upgrade our crypto to some more complex curve space, because all of them will be vulnerable to Shor's algorithm.

The mathematics behind Shor's algorithm is actually a really interesting read if you enjoy pure, abstract algebra and topology.