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Mathematics of Computation

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Computing isomorphisms and embeddings of finite fields

Authors: Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori and Éric Schost
Journal: Math. Comp. 88 (2019), 1391-1426
MSC (2010): Primary 13P05, 68W30
Published electronically: June 19, 2018
MathSciNet review: 3904150
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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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Additional Information

Ludovic Brieulle
Affiliation: Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay

Luca De Feo
Affiliation: Laboratoire de Mathématiques de Versailles, UVSQ, CNRS & Inria, Université Paris-Saclay

Javad Doliskani
Affiliation: Institute for Quantum Computing, University of Waterloo

Jean-Pierre Flori
Affiliation: Agence nationale de la sécurité des systèmes d’information

Éric Schost
Affiliation: Cheriton School of Computer Science, University of Waterloo

Received by editor(s): May 5, 2017
Received by editor(s) in revised form: July 3, 2017, and December 18, 2017
Published electronically: June 19, 2018
Article copyright: © Copyright 2018 Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori, and Éric Schost