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

I wasn't able to find any solution for 32-bit unsigned integers but I've found some solutions for 2-bit integers which I was trying to use in my Prolog program. One of my solutions (which uses exponentiation and modulo) is described in this StackOverflow question and the others (some without exponentiation, pure algebra) can be found in this code repository on Github: see different xor0 and o_xor0 implementations.

The nicest xor represention for 2-bit uints seems to be: xor(A,B) = (A + B*((-1)^A)) mod 4.

Solution with +,-,*,/ expressed as Excel formula (where cells from A2 to A5 and cells from B1 to E1 contain numbers 0-4) to be inserted in cells from A2 to E5:

(1-$A2)*(2-$A2)*(3-$A2)*($A2+B$1)/6 - $A2*(1-$A2)*(3-$A2)*($A2+B$1)/2 + $A2*(1-$A2)*(2-$A2)*($A2-B$1)/6 + $A2*(2-$A2)*(3-$A2)*($A2-B$1)/2 - B$1*(1-B$1)*(3-B$1)*$A2*(3-$A2)*(6-4*$A2)/2 + B$1*(1-B$1)*(2-B$1)*$A2*($A2-3)*(6-4*$A2)/6

Answer 2

I think this relation might help in answering your question

A + B = (A  XOR  B ) + 2*(A.B)