I was reading about a cipher called Speck, which defines a system of equations using Addition Mod $2^{32}$ ($\boxplus$), Bit Rotation, and XOR.
If we pretend that the additions were taken over $\mathbb{Z}/n\mathbb{Z}$ for $n = 2^{32}-1$, since there are a small number of additions in a complete application of the cipher, we can fudge whether $a \boxplus b = a + b$ or $a \boxplus b = a + b - 1$. And in such a setting rotation becomes multiplication by 2.
But how do you deal with the XORs? Is there a tidy way to represent them $\mathbb{Z}/n\mathbb{Z}$?
This might be a stupid idea, but I was thinking about $F = \mathbb{Q}[\omega]$ where $\omega$ is a complex $n$th root of unity. Addition in this setting becomes multiplication, multiplying by two becomes squares, and I have a hunch about xor. Since $n$ is the product of Fermat primes, I think that means that $F$ is a bunch of quadratic extensions of $\mathbb{Q}$.
Do the member of the Galois group of $F$ relate in any way to the application of XOR by a constant value?