I just found out about finite fields because of AES but I think it's may have illuminated something about the Pollard Rho Brent prime decomposition algorithm for me:
(((x*x)%N)+c)%N
Where N=pq and p and q are primes, and x & c are integers.
Would I be correct in saying that this is just a squaring and an addition performed on a finite field? I'm unsure because the video I'm watching is telling me that a finite field has to have a prime or prime power number of elements but the value of N seems to conform to neither.
Working $\mod pq$ is not a finite field because $p$ and $q$ do not have multiplicative inverses. There is no number you can multiply them by and get $1$. You are working in a finite ring, which means you have addition, subtraction, and multiplication with multiplication distributing over addition. It is very much like the integers. Division works sometimes, in this case for numbers coprime to both $p$ and $q$.