Regarding [2]: Xor and And form a basis of all logical operations, while Xor on its own (or And on its own) do not. In GF(2) multiplication is xor and addition

Regarding [2]: Xor and And form a basis of all logical operations, while Xor on its own (or And on its own) do not. In GF(2) multiplication is xor and addition is and. So addition and multiplication are enought to build all possible circuits. Another way to look at it is, in homomorphic encryption you have the problem that in an operation both operands are encrypted. Therefore repeated addition based on one operand is not an option.