On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field $\F_q$ of $q$ elements from queries to an oracle that for a given $x\in \F_q$ 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.

💡 Research Summary

The paper studies the “Hidden Shifted Power Problem” (HSPP) over a finite field 𝔽_q with q elements. An oracle O_{e,s} takes an input x∈𝔽_q and returns (x+s)^e, where e divides q−1. The goal is to recover the hidden shift s using as few oracle queries as possible. The naïve approach interpolates a degree‑e polynomial using e+1 queries, which is optimal in the worst case but can be costly when e is large, especially in cryptographic settings where each oracle call is expensive.

The authors introduce a new methodology based on additive combinatorics and analytic number theory. They define a polynomial \