Let $GF(p) = ({\mathbb Z}_p, +, \times)$ be the Galois field where $p>2$ is prime and let $$ H=\{1,2,\cdots, \frac{p-1}{2}\}.$$
I need an algorithm (subexponential in terms of $\log_2 p$) that computes at least one element inside the following expression (or, indicate that it is empty) $$ \bigcap_{i=1}^n a_i H $$ where $a_1, \cdots, a_n\in {\mathbb Z}_p\setminus \{0\}$ are given as inputs and $aH=\{a\times x\mid x\in H\}$. I believe that this problem is NP-hard. Can someone give any hints, links, maybe related problems.