# Meaning of difference Δ delta

I tried to understand some crypto text but I am new to this.

Does this:

Let $$X,X'$$ be $$\ell$$-bit values, and $$\Delta X=X\oplus X'$$

means that $$X$$ and $$X'$$ have $$\ell$$ bits and $$\Delta X=X-X'$$ or something else

means that $$X$$ and $$X'$$ have $$\ell$$ bits[...]?
means that [...] $$\Delta X=X-X'$$[...]?
Yes and no. Fundamentally what $$x-y$$ means is $$x+(-y)$$, but for bit-strings over $$\mathbb F_2$$ this becomes $$x+(-y)\bmod 2$$ for each bit, and $$-y\equiv y\pmod 2$$, i.e. $$y$$ and $$-y$$ behave the same. As this means that addition is just XOR, we write $$x\oplus y$$ to indicate this.
Also note that $$\Delta X=X\oplus X'$$ is the bitstring that has a bit set iff $$X$$ and $$X'$$ differ in that bit position and so it's the "difference" ($$\Delta$$) between the strings.
• You also might mention that this $\Delta$ is used in Differential Cryptanalysis and $\oplus$ is the natural choice for the $\Delta$ since the key is $\oplus$'ed with the current state. This doesn't mean that $\Delta$ is always useful with $\oplus$, however, couldn't see some other choices. Mar 8 '20 at 19:05