# Values of rational functions in small subgroups of finite fields and the identity testing problem from powers

## TL;DR

This paper investigates the behavior of rational functions within small subgroups of finite fields and applies these insights to improve understanding of the identity testing problem for polynomials over field extensions.

## Contribution

It provides new lower bounds on subgroup sizes containing rational function images and applies these bounds to the identity testing problem in finite fields.

## Key findings

- Established lower bounds on subgroup sizes for rational function images.
- Applied bounds to analyze the complexity of identity testing algorithms.
- Enhanced understanding of polynomial identity testing in high-degree field extensions.

## Abstract

Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field~$\mathbb{F}_{q^n}$ considered as a linear space over a subfield $\mathbb{F}_q$. We apply this to the recently introduced algorithmic problem of identity testing of "hidden" polynomials $f$ and $g$ over a high degree extension of a finite field, given oracle access to $f(x)^e$ and $g(x)^e$