# Computing isomorphisms and embeddings of finite fields

## TL;DR

This paper reviews, improves, and implements algorithms for computing isomorphisms and embeddings of finite fields, demonstrating superior performance over existing software through detailed complexity analysis and experiments.

## Contribution

It introduces improved algorithms for finite field isomorphism and embedding problems, with comprehensive complexity analysis and open-source implementation.

## Key findings

- New algorithms outperform existing methods in efficiency.
- Detailed complexity analysis confirms competitiveness of proposed variants.
- Experimental results show consistent performance improvements over current software.

## Abstract

Let $\mathbb{F}_q$ be a finite field. Given two irreducible polynomials $f,g$ over $\mathbb{F}_q$, with $\mathrm{deg} f$ dividing $\mathrm{deg} g$, the finite field embedding problem asks to compute an explicit description of a field embedding of $\mathbb{F}_q[X]/f(X)$ into $\mathbb{F}_q[Y]/g(Y)$. When $\mathrm{deg} f = \mathrm{deg} g$, this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better. We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.