A cute theorem involving xor
Define [a..b] = \set{x|a\leq x\leq b, x\in \mathbb{N} }, and k\oplus [a..b] = \set{k\oplus x| x\in [a..b]}, where \oplus is the bitwise xor function. We want to know something about k\oplus [a..b].
x-y \leq x\oplus y \leq x+y
x\oplus y \leq x+y is easy to see as \oplus is binary addition without carries. Assume x\oplus y < x-y for some x and y, then [ x = x\oplus(y\oplus y) = (x\oplus y) \oplus y < (x-y)\oplus y\leq (x-y)+y = x ] A contradiction. Therefore x-y \leq x\oplus y.
\oplus, binary addition without carries, and binary subtraction without borrows are the same operation. The lemma is trivial by notice that equivalence.
If f:\mathbb{N}\to \mathbb{N} a surjection such that x-n\leq f(x)\leq x+m, then
- [a+m..b-n] \subseteq f([a..b])
- [0..b-n] \subseteq f([b])
f is a surjection implies all values in any integer interval gets taken. f^{-1}(y)=\min(\{x|f(x)=y\}) [ x-n\leq f(x)\leq x+m \implies y+m\leq f^{-1}(y)\leq y-n. ]
- If y\in [a+m..b-n], f^{-1}(y) \in [(a+m)-m.. (b-n)+n] = [a..b].
- If y\in [0..b-n], f^{-1}(y) \in [\max(0-m,0).. (b-n)+n] = [0..b].
Let f_k(x) = k\oplus x, noting that f_k is a bijection, we derive the result we want for xor.
- [a+k..b-k] \subseteq k\oplus [a..b].
- [0..b-k] \subseteq k\oplus [0..b].