The existence of partial homomorphic additive encryption with bit-wise XOR operation
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.