I know that $XOR$ is equivalent to modular addition in the field $\mathbb{F}_2 = \{0,1\}$ (is it right?), and thus should satisfy the following property of a bilinear form:

$$\oplus(u+v,w) = \oplus(u,w) + \oplus(v,w)$$

My doubt is: how has the $+$ in the previous formula to be interpreted? Because if I interprete it like the same $XOR$ (even if looking at the formula it seems not to make much sense), applying the property I think I would have:

$$(u\oplus v) \oplus w = (u \oplus w) \oplus (v \oplus w) = u \oplus v $$

Even though I was expecting a result like:

$$ u \oplus v \oplus w$$

If the interpretation of the $+$ like the $\oplus$ is not the right interpretation, how has it to be interpreted, even considering that the normal $+$ and the $\oplus$ coincide in $\mathbb{F}_2 = \{0,1\}$?

At the end, what's the meaning of that property?


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 fundamental properties of a field; since $GF(2)$ is a field (with $\oplus$ and $\otimes$ as the field addition and field multiplication), this holds.

And, in $GF(2)$, $\otimes$ is equivalent to bitwise $AND$

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  • $\begingroup$ so $\oplus$ is not bilinear? $\endgroup$ – Alessio Martorana Jul 12 '16 at 10:36
  • $\begingroup$ No, $\otimes$ is blinear (over $\oplus$) $\endgroup$ – poncho Jul 12 '16 at 12:06
  • $\begingroup$ But is $\oplus$ a linear operation? $\endgroup$ – Alessio Martorana Jul 13 '16 at 1:43
  • $\begingroup$ @AlessioMartorana: yes, it is (over the group $GF(2)$, which is a bit tautological when you think about it...) $\endgroup$ – poncho Jul 13 '16 at 3:24
  • $\begingroup$ I was asking precisely that: what is the difference between being linear in $GF(2)$ or in, for example, $\mathbb{Z}_n$ ? What's the meaning of this? And why $\oplus$ is linear in $\mathbb{F}_2$ and not in $\mathbb{Z}_n$, what does not happen? $\endgroup$ – Alessio Martorana Jul 13 '16 at 10:53

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