I'm interested in understanding the theory behind and around elliptic curves. I have a basic idea what an elliptic curve is in a practical sense and how to use a provided curve for cryptographic purposes, but I'm struggling to understand several papers about generating curves with certain properties.
My ultimate goal is to be able to generate elliptic curves in different prime fields with whatever properties are required to plausibly support the exponential-hardness decisional Diffie-Hellman problem.
Some papers I'm trying to read are:
- Counting points on elliptic curves over finite fields
- Efficient ephemeral elliptic curve cryptographic keys
- Accelerating the CM Method
I've also tried to read many other papers on the on generating elliptic curves, point counting, and calculating various properties of elliptic curves, and I struggle with the same problem in all of them:
I'm not at all knowledgeable about group theory, class polynomials of different types, endomorphisms, endomorphism rings, imaginary quadratic orders, the discriminat of different constructs, what $h(D)$ is in the context of the class polynomial $H_D(X)$, and many other things that seem required to understand such papers.
What are some introductory papers, books, or other resources that could familiarize me with these (and other) building blocks I need in order to understand these more advanced papers?
Edit 1: I understand the building blocks along with elliptic curves themselves span several large areas of mathematics and that it will take serious effort and significant time to learn. This roadblock is something I keep running into when trying to read interesting-sounding papers about other topics I'm interested in, so I'm willing to invest the effort and time required to learn and understand these topics.