My question refers to the paper "Fully homomorphic encryption modulo Fermat numbers" by Antoine Joux. On page 3, the author describes a basic concept of the system:
As many FHE systems, we deal with noisy messages. In our case, the high bits of each block are used to hold significant bits, while the low bits contain noise. A fundamental identity that makes the system work is that given two bits x and y , we have: x + y = 2(x ∧ y) + (x ⊕ y). Thus, if we can add the values of two bits as integers, or even as integers modulo 4, we are simultaneously computing an AND and a XOR gate.
I don't understand the difference between "integers" and "integers modulo 4". If we have only two variables which hold a single bit, the sum of both can be a maximum of 2. So what is the necessity to use a modulo 4 ring? And why not modulo 3, modulo 8 or any other number?