Current mathematics theory used in cryptography/coding theory What are the mainstream techniques borrowed from algebraic geometry (or some other branch of mathematics) which are currently used in cryptography/coding theory? I've only heard about a small subset of elliptic curves and hermitian curves. I've also heard about research of hyperelliptic curves but I don't know if some software has already implemented it.

Could you mention some other branches of mathematics being strongly used in cryptography/coding theory and its state? (mainstream/in research)
Finite fields - which is a branch of algebra - is a must. It is, in some way, used in almost all types of cryptographic algorithms.

Also, you need some sort of basic programming ability since you will need to calculate time and space complexity of cryptographic algorithms. An "Algorithms" course  taken from a CS  department would be very useful.

My advice is, you can read the  first 2-3 chapters of Rudolf Lidl' s Finite Fields book. You don't need to understand every little detail in the book, just read that part and get the general idea, and that would be enough for the beginning.

But the bible for the beginners in cryptography is definitely, "Introduction to Cryptography" by Washington and Trappe. It is the most basic book amongst all in addition to being very  comprehensive. It  has a solution manual and nice exercises that you will enjoy solving. I personally learned a lot from it. Trappe covers almost every basic thing that you need to know as a beginner.