BALTIC-WAY 2017 Problem 18

Let $p>3$ be a prime and let $a_{1}, a_{2}, \ldots, a_{\frac{p-1}{2}}$ be a permutation of $1,2, \ldots, \frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_{1}, a_{2}, \ldots, a_{\frac{p-1}{2}}$ if for all $i, j \in\left\{1,2, \ldots, \frac{p-1}{2}\right\}$ with $i \neq j$ the residue of $a_{i} a_{j}$ modulo $p$ is known?