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Based on the additional details in the comments, it seems like your question is: given $c_1=a\oplus d$ and $c_2=b\oplus d$, can we get $(a+b)\oplus d$. Where $a,b,d\in\mathbb{Z}_p$, $+$ is addition modulo $p$, and $\oplus$ is a bitwise XOR of the values, then taken modulo $p$. Or put another way, is there an operation $\boxplus$, such that $(a\oplus ... 0 η is the bit-length of the secret key (which is the hidden approximate-gcd of all the public-key integers), SO u can calculate given functions and P should be the key that will be used and it should be between [2^η−1,2^η) Hope this answers your question 4 It's actually straight-forward; we'll assume that all the inputs are either encrypted versions of 0, or encrypted version of 1; then: We can replace an AND gate with just an FHE multiplication of the two inputs: $$AND(x,y) = x*y$$ Where$*\$ is our Homeomorpic multiplcation operation. This obviously evaluates to an encrypted 1 if both of the inputs are ...