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
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means that $X$ and $X'$ have $\ell$ bits[...]?
Yes.
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.
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$\begingroup$ Thank you very much! Now I understand finally:) $\endgroup$ – blissfull-999 Mar 8 '20 at 13:10
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1$\begingroup$ 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. $\endgroup$ – kelalaka Mar 8 '20 at 19:05
