Figuring out which grad schools/what department to apply to

[PhDorBust]

So I am a final year math student and there are two questions here,

1) I am a math student but all of my research has been in CS. If I do pursue a CS degree it will be in something mathematical like cryptography, but I really have no clue. Should I be applying to Math or CS departments? Does it matter in the long run?

2) How do you know what range of schools is appropriate to be applying to? I have 4.0 gpa, REUs at Cornell and UCLA, 1 publication from other research at home institution, top percentile on GRE, but I come from a poor undergrad school. That is, given say 10 applications, how would you distribute them between top 20 schools, top 20-50 schools, etc?

I know rank-centric sorting may not be most accurate measure, but I don't see another easy way to partition good vs not-as-good.

 

[Bourbaki1123]

What was the subject matter of your two REUs? Do you have any particular research interests that you could articulate in any detail?

What sort of cryptography are you interested in? There has been quite a stir surrounding homomorphic encryption schemes. There is also algebraic cryptanalysis to consider, which has much more of a discrete mathematics feel (lots of finding complexity bounds for algorithms on polynomial rings with finite ground fields, for instance).