Is there any possibility to recover A in "A & B = C" with given B and C?

Question

I would like to ask: with A, B, and C are any binary number. After getting C = A & B (& is AND operator), is there any possibility to recover A from B and C?

I know that the information of A will be lost through the operation. Can we form a function like B <...> C = A, and how complexity it can be?

For example:

A = 0011
B = 1010
C = A & B = 0010

The 2nd bit of C is 1, i.e. 2nd bit of A and B must be 1. However, the other bits lack information to be recovered.

Thank you in advance.

Answer 1

You can't recover A, but you can write A = (X & ~B) ^ C. Here, X can be anything (and it gives all the A's).

Of course this will work only for B and C such that C & ~B == 0.

This is a parametrized solution. Example in python

>>> A = 32776466
>>> B = 89773888
>>> C = A & B
>>> C
22020352
>>> X = 1234567890    # arbitrary value
>>> U = (X & ~B) ^ C
>>> U
1238761874
>>> U & B     # same result as A & B
22020352