I am having trouble with the following section "Convolution Polynomial Rings" in "Introduction to Mathematical Cryptography" by Hoffstein et al..
I do not get what is going on at the last equation sign, could you please explain this to me?
Remark: Here $R := \mathbb{Z}{[x]} / (x^N-1)$
Note that if $i + j = k$ then $(i + j) - k = 0 = 0N$.
If $i + j = k + N$ then $(i+j) - k = N$
By definition: If $a - b = cN$ for some integer c, then $a \equiv b$ (mod $N$)
Also note that $i$ and $j$ only goes up to $N-1$ for each of them so $i + j - k \leq 2N - 2 \neq 2N$ so really we can only have $c=0,1$. Increasing the index up to $N-1$ on the 2nd summation does not change this.
So we can combine the summations and just use $(i + j) \equiv k$ (mod $N$) as a qualifier for the inner sum.