On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field ${\text{F}}_q$ of $q$ elements from queries to an oracle that for a given $x\in {\text{F}}_q$ returns $(x+s{)}^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.