Cryptography XOR Question Recently I encountered an interview question related to XOR and I have no idea how to figure it out.
The question is as below:

The algorithm is: $A\oplus B$ (operation) $C = D$

We have the value of $A$, $B$ and $D$. Please figure out the value of $C$ and the (operation). The $C$ is being dump as per the interviewer.

Is there a way to figure out the value of $C$ and the operation as in the question?
First of all I'll have to disagree with fkraiem that there are many possibilities. In fact there are exactly four. I'm going to assume that $A$, $B$, $C$, and $D$ are bits. (Or equivalently I'm assuming that they are bit-strings and the operation we are looking for a is a bitwise boolean operation.)
Then I'll make my life a bit easier and define $E=A\oplus B$, since it does not matter what $E$ actually is.

Now, there are exactly 16 possible binary boolean operations. Not all of those actually have their own name, but they technically do exist and we could come up with a symbol for them. Assuming that we are supposed to find an operation and a description of $C$ in terms of $D$ and $E$ such that the equation is satisfied for all possible values of $E$ and $D$.

Then this leaves us with exactly 4 possibilities.

Namely


XOR as the operator and $C=E\oplus D$.
XNOR as the operator and $C= \neg(E\oplus D)$.
The operator that always evaluates to the second value and $C=D$.
The operator that always evaluates to the negation of the second value and $C=\neg D$.


The reason for this can be seen in the following table. For an operation to be viable, we need that it allows us to force any value of $D$ irrespective the value of $E$. This means we require that in the following table the operation has both a $0$ and a $1$ in each pair of columns in the following table. Otherwise there is always at least one combination of values for $E$ and $D$ that makes the equation unsatisfiable. It is easy to see that there are only 4 viable operations.