My background is computer scientist. I have done applied cryptography research for a while. Currently, I'm working on Elliptic curve cryptography.
However, When it comes to improving ECC research, it requires a lot of mathematical knowledge. For instance, if I want to introduce a new curve (like when Bernstein, Birkner, Joye, Lange, and Peters introduced Twisted Edwards curve in 2008) and evaluate its security, or trying to solve Elliptic Curve Discrete Log Problem (ECDLP), what math field(s) should I study?