Mathematical (Arithmetic) representation of XOR

Question

I have spent the last 5 hours searching for an answer. Even though I have found many answers they have not helped in any way.

What I am basically looking for is a mathem

Answer 1

Given that (a-b)(a-b) quite obviously computes xor for a single bit, you could construct a function with the floor or mod arithmetic operators to split the bits out, then xor them, then sum to recombine. (a-b)(a-b) = a² -2·a·b + b² so one bit of xor gives a polynomial with 3 terms.

Without floor or mod, the different bits interfere with each other, so you're stuck with looking at a solution which is a polynomial interpolation treating the input a,b as a single value: a xor b = g(a · 2³² + b)

The polynomial has 2⁶⁴-1 terms, though will be symmetric in a and b as xor is commutative so you only have to calculate half of the coefficients. I don't have the space to write it out for you.