Finite field arithmetic is basically what you would get if you were to reduce the integers to a set of finite size, but keep the arithmetic rules similar to wha

Finite field arithmetic is basically what you would get if you were to reduce the integers to a set of finite size, but keep the arithmetic rules similar to what they were before. Using an infinite set like the integers is a bad idea because you can't have a uniform distribution over all the integers, and hence some members will be more likely than others, which leaks information. The special case of finite-field arithmetic we usually care about is GF(p), which is when the integers start at 0 and wrap around at p, where p is a prime number. We care about this because when p is prime, then we can ensure numbers are uniformly random in the range 0..p-1, which is something we need in order to guarantee information-theoretic security.