ROAD MAP

Step 1: Refresh on Number Theory Basics Start with the fundamentals of number theory. Focus on concepts like modular arithmetic, prime numbers (even though you know them, it helps to see them from a number theory angle), greatest common divisors, and the Chinese Remainder Theorem. A good number theory book or an online course could take you through this in maybe a month or so.

Step 2: Dive into Abstract Algebra Next, you’ll want to get a handle on some abstract algebra, especially group theory and a bit of field theory. This will help you understand the structure behind things like elliptic curves and finite fields. You don’t need to become a full-blown algebraist—just get comfortable with the concepts that are commonly used in cryptography. This might be another couple of months of part-time study.

Step 3: Elliptic Curves and Finite Fields Once you have the algebra background, you can dive into the math of elliptic curves more deeply—understanding the equations, the group law on the curve, and how the finite fields work with them. This is where you really connect the math to the cryptographic applications you already know. Another month or so of focused learning could get you pretty far here.

Step 4: Dive Into Cryptographic Proofs and Security Models Finally, once you have those math foundations, you can look at the more theoretical side of cryptography—how we prove that certain algorithms are secure, the mathematical reasoning behind cryptographic hardness assumptions, and so on. At this point, you're basically polishing and deepening your understanding, and you can do this alongside whatever practical work you're already doing.