Rewrite $\left\vert\sum_{k\in B} \exp\left(\frac{2\pi ijk}{p} \right)\right\vert^2$, where $p$ is a prime $\ge 3$ and $B \subset \{1,2,...,p-1\}$.

Question

If $p$ is a prime number greater than or equal to $3$ and $B$ is a subset of $\{1,2,...,p-1\}$, why is it true that $$\left\vert\sum_{k\in B} \exp\left(\frac{2\pi ijk}{p} \right)\right\vert^2 = \sum_{k_1,k_2 \in B} \exp \left( \frac{2\pi ij (k_1-k_2)}{p}\right)?$$

Do we use the following fact? Let $e(t)$ denote $\exp\{2\pi i t\}$. For integers $m$ and $j$, it is true that $\sum_{k=0}^{m-1} e(jk/m) = 0$ if $m$ does not divide $j$; and $m$ if $m$ does divide $j$.

Answer 1

Just expand the LHS as the product $S S^{\ast}$