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Well, as the author of the answer you cited, I can't much improve on tylo's answer. However, perhaps a couple of examples will be an useful supplement. First, note that for $a,b \in \mathbb{Z}_4$: $$a \oplus b \equiv a + b + 2ab \pmod 4$$ Where the $\oplus$ on the left side means XOR and on the right side are addition and multiplication modulo 4. In ...

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Sicne your question is refering to an answer on this site, I will not quote the entire answer here. But the crucial point is there in the last paragraph: If none of that made any sense to you, that's OK. You just need to read up about Abstract Algebra, and Fields, Rings and Groups. It's a fascinating and beautiful area of mathematics, and much of ...

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Actually, as you note, XOR doesn't satisfy the bilinear condition. What does is multiplication $\otimes$ in $\mathbb{F}_2$, that is: $$\otimes(u \oplus v, w) = \otimes(u,w) \oplus \otimes(v,w)$$ or, as more traditionally expressed (using $+$ and $\times$ as the field operations): $$(u+v) \times w = (u \times w) + (v \times w)$$ This is one of the ...

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