The existence of partial homomorphic additive encryption with bit-wise XOR operation According to  Protocol A that was presented in Section 3.1  paper entitled "Some Efficient Solutions to Yao’s Millionaire Problem" (2013). [1]

In that protocol they used an assumption that there is an encryption function that has the homomorphism property with respect to both additions (over some finite field) and bitwise XOR operations.

Can someone please send a reference to scheme that uphold the requirements?

My current candidate is to combine the answer in [2] and micali-goldwaser encryption [3].

But I don't know how (and if it possible) the scheme could work  and overcome:
(1) $N$ in GM  cannot be a $Z_{2^q}$ as it's a multiplication of two primes.
and
(2) Is it possible to work with 2-complement binary representation ( To perform the subtraction operation).

[1]- https://arxiv.org/pdf/1310.8063v1.pdf

[2]- Homomorphic encryption based on XOR

[3]- https://en.wikipedia.org/wiki/Goldwasser%E2%80%93Micali_cryptosystem
Assuming the finite field is of odd characteristic (that is, $GF(p^k)$ for some prime $p &gt; 2$, then such an encryption operation would be FHE; and would be much bigger news than just only solving the Millionaire problem.

With such an encryption method, given $E(a)$, $E(b)$ and a constant $c$, we can compute $E(c)$,$E(a \oplus b)$, $E(a + b)$ and $E(c \times a) = E(\underbrace{a + a + ... + a}_{\text{c times}})$ (where the addition and multiplication are within $GF(p^k)$,   If we can't compute $E(c)$ directly, we can just place $E(c)$ within the public key (assuming that the range of possible $c$'s is limited).

So, assuming $a, b$ are restricted to $\{0, 1\}$, we can compute the NAND operation $E(\overline{a \wedge b}) = E( 1 \oplus (k \times ((a + b) \oplus a \oplus b)))\ $ for the constant $k = 2^{-1} \pmod p$.  It is straightfoward to see that this function gives $E(1)$ if either of $a, b$ is 0, and $E(0)$ otherwise.

With this encrypted NAND function, we can construct any circuit.