On the Hidden Shifted Power Problem

TL;DR

This paper introduces more efficient algorithms for recovering a hidden element in a finite field using fewer oracle queries by applying additive combinatorics and number theory techniques.

Contribution

It presents novel algorithms that improve query efficiency for the Hidden Shifted Power Problem over previous methods.

Findings

Reduced number of oracle queries compared to naive methods

Algorithms based on additive combinatorics and number theory

Enhanced efficiency in recovering the hidden element

Abstract

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.