Is it a requirement to understand mathematics when implementing (or breaking) cryptography?
I would say there are three general areas of necessary expertise for most crypto-related jobs:


Knowledge of primitives and their use cases.
Knowledge of protocols and understanding how to reason about their security.
Deep and abiding understanding of how incredibly stupid people are, including oneself.  


The most that knowing the math is going to do for you practically, unless you are in academia/the NSA designing algorithms, is tell you "time to stop using RSA" on the day that you hear that there is a polynomial time factoring algorithm.