https://github.com/defeo/ffisom

A research project on isomorphisms of finite fields
https://github.com/defeo/ffisom

algorithms finite-fields isomorphism research-paper

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A research project on isomorphisms of finite fields

• Host: GitHub
• URL: https://github.com/defeo/ffisom
• Owner: defeo
• Created: 2013-07-12T08:11:56.000Z (over 11 years ago)
• Default Branch: master
• Last Pushed: 2018-06-15T03:23:45.000Z (over 6 years ago)
• Last Synced: 2024-04-15T03:04:10.987Z (7 months ago)
• Topics: algorithms, finite-fields, isomorphism, research-paper
• Language: Jupyter Notebook
• Homepage: https://arxiv.org/abs/1705.01221
• Size: 36.1 MB
• Stars: 16
• Watchers: 12
• Forks: 5
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# Computing isomorphisms and embeddings of finite fields

*a research project in Computer Algebra*

by [Ludovic Brieulle](https://github.com/brieulle/), [Luca De

Feo](http://defeo.lu/), [Javad

Doliskani](https://github.com/javad-doliskani), [Jean-Pierre

Flori](https://github.com/jpf-anssi/) and [Éric

Schost](http://www.csd.uwo.ca/~eschost/).

#### Abstract

Let *𝔽q* be a finite field.  Given two irreducible

polynomials *f*, *g* over *𝔽q*, with deg *f* dividing deg

*g*, the finite field embedding problem asks to compute an explicit

description of a field embedding of *𝔽q[X]/f(X)* into

*𝔽q[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 open source.  Our experiments show that our new variants

consistently outperform available software.

## Research paper

The research paper has been submitted for publication in [AMS

Mathematics of

Computation](http://www.ams.org/publications/journals/journalsframework/mcom).

A preprint version is available at .

The LaTeX sources to the paper can be found in the [`paper`](paper)

folder.  They are subject to the copying restriction enforced by the

editor.

## Software [![launch binder](https://mybinder.org/badge.svg)](https://mybinder.org/v2/gh/defeo/ffisom/master?filepath=notebooks)

The [`implementation`](implementation) folder contains source code for

the algorithms implemented by the project. See the folder `README.md`

for instructions on compiling and running the software on your own

machine.

The [`notebooks`](notebooks) folder contains Jupyter notebooks showing

sample usage of the software and benchmarks. Notebooks can be

statically viewed on GitHub, or **executed** in

[Binder](https://mybinder.org/v2/gh/defeo/ffisom/master?filepath=notebooks),

thus saving you the time to compile and install the software.

All source code is distributed under the MIT license.

## Related material

The [`misc`](misc) folder contains material related to the project,

such as

- conference posters,

- presentation slides,

- experimental data gathered on the conjecture formulated in the

  paper.

## Contributing

We do not accept pull requests, but if you see any problem with this

repository, plese open an issue.