I'm curious about the extent of usage of mathematics in research in cryptography.
I'm considering doing a Ph.D in cryptography, and I'm particularly interested in elliptic curve cryptography. However I can't tell if actual research in cryptography would involve studying theoretical structure of elliptic curves. For example, I'd be more interested if I am to compute j-invariant or Selmer group at some point, and I'd be less interested if most of my job is about scrutinizing computer systems.
Could anyone familiar with research in cryptography help me out? Thank you in advance.