XOR and bilinear form property 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 $AND$