Q:

The sign of the correlation in linear approximation

${\text{Let }}g:{\{ 0,1\} ^n} \to {\{ 0,1\} ^n}$ be a round function of an $n$-bit iterated block cipher and let $K \in {\{ 0,1\} ^n}$ be a fixed round-key, and denote ${g_K}(x) = g(x \oplus K).$ Show that $${\text{cor}}(u \cdot x,v \cdot {g_K}(x)) = {( - 1)^{u \cdot K}}{\text{cor}}(u \cdot x,v \cdot g(x))$$

A:

Since $c(f,g)=\sum_x (-1)^{f(x)\oplus g(x)}$, we can write  $$c(u\cdot x, v \cdot g(x\oplus K))=\sum_x (-1)^{u\cdot x \oplus g(x\oplus K)}$$ where the sum ranges over all binary strings of length $n$ which is a vector space over the binary field. Now let $y=x\oplus K$ and note that as $x$ ranges over this vector space so does $y$. Therefore we can rewrite the same correlation as 
$$c(u\cdot x, v \cdot g(x\oplus K))=\sum_y (-1)^{u\cdot (K \oplus y) \oplus g(y)}=(-1)^{u\cdot K}\sum_y (-1)^{u\cdot  y \oplus g(y)}.$$
The last sum is just $c(u\cdot x, v \cdot g(x))$ thus giving the claimed result.
This is also related to a shift property of the Walsh Hadamard transform which is after all a Fourier transform.

