Meaning of "integers modulo 4 " in "Fully homomorphic encryption modulo Fermat numbers" scheme 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?
"Integers modulo 4" is usually the finite ring $(\Bbb Z_4,+,*)$ which internal laws are 

+ | 0 1 2 3     * | 0 1 2 3
--+--------     --+--------
0 | 0 1 2 3     0 | 0 0 0 0
1 | 1 2 3 0     1 | 0 1 2 3
2 | 2 3 0 1     2 | 0 2 0 2
3 | 3 0 1 2     3 | 0 3 2 1


but here, the question's citation only deals with the finite group $(\Bbb Z_4,+)$.




  difference between "integers" and "integers modulo 4"


In the later, everything is reduced to four values, which can be expressed on two bits, the two lower bits of normal integers.