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.

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Where would a PhD in cryptography program be housed? A math department? A computer science department? This would likely determine, at least partially, what type of program you would be entering. Every program will be slightly different. You would have to look into each individually. My guess is that there are very very few places that offer a PhD specifically focused on cryptography.

In my experience in math, most PhD programs in cryptography are in a computer science department. Algebraic number theory would more be the math subject one would study.

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