Computing isomorphisms and embeddings of finite fields

Brieulle, Ludovic; De Feo, Luca; Doliskani, Javad; Flori, Jean-Pierre; Schost, Éric

doi:10.1090/mcom/3363

Computing isomorphisms and embeddings of finite fields
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by Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori and Éric Schost;

Math. Comp. 88 (2019), 1391-1426

DOI: https://doi.org/10.1090/mcom/3363

Published electronically: June 19, 2018

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Abstract:

Let $\mathbb {F}_q$ be a finite field. Given two irreducible polynomials $f,g$ over $\mathbb {F}_q$, with $\deg f$ dividing $\deg g$, 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 $\deg f = \deg g$, 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 an open source. Our experiments show that our new variants consistently outperform available software.

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