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Of course, it is possible. It is possible to take a public key encryption algorithm which takes the private key and the known ciphertext into the known plaintext, convert that into a set of equations in $GF(2)$ with the private key as unknown variables, and solve for those unknown variables. Alternatively, you can take the key generation algorithm, which ...
There is of course but because of carry bits there will be data-dependent "nonlinear" terms. If I do it for 2 bits you can get the idea. It gets unwieldy but you can easily write code to do it for longer bitlengths. The list below is an XOR table expressed as integers:  \begin{array}{c|ccc} \oplus & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & ...