egcd

Pure-Python extended Euclidean algorithm implementation that accepts any number of integer arguments.

Installation and Usage

This library is available as a package on PyPI:

python -m pip install egcd

The library can be imported in the usual way:

from egcd import egcd

Examples

The function egcd is a pure-Python implementation of the extended Euclidean algorithm that can be viewed as an expansion of the functionality and interface of the built-in math.gcd function. When it is supplied two integer arguments a and b, it returns a tuple of the form (g, s, t) where the three integers in the tuple satisfy the identity (a * s) + (b * t) == g:

>>> egcd(1, 1)
(1, 0, 1)
>>> egcd(12, 8)
(4, 1, -1)
>>> egcd(23894798501898, 23948178468116)
(2, 2437250447493, -2431817869532)
>>> egcd(pow(2, 50), pow(3, 50))
(1, -260414429242905345185687, 408415383037561)

However, any number of integer arguments can be supplied. When no arguments are supplied, the result is (0,) (just as the expression math.gcd() evaluates to 0 in Python 3.9 and higher). In all other cases, the result contains the greatest common divisor of all the supplied integers and the coefficients of the generalized form of the associated identity:

>>> egcd(2, 4, 3, 9)
(1, -1, 0, 1, 0)
>>> 1 == ((-1) * 2) + (0 * 4) + (1 * 3) + (0 * 9)
1

A succinct way to extract the greatest common divisor and the coefficients is to take advantage of Python's support for iterable unpacking via the asterisk notation:

>>> bs = (26, 16, 34)
>>> (g, *cs) = egcd(*bs)
>>> (g, cs)
(2, [-3, 5, 0])
>>> g == sum(c * b for (c, b) in zip(cs, bs))
True

Development

All installation and development dependencies are fully specified in pyproject.toml. The project.optional-dependencies object is used to specify optional requirements for various development tasks. This makes it possible to specify additional options (such as docs, lint, and so on) when performing installation using pip:

python -m pip install ".[docs,lint]"

Documentation

The documentation can be generated automatically from the source files using Sphinx:

python -m pip install ".[docs]"
cd docs
sphinx-apidoc -f -E --templatedir=_templates -o _source .. && make html

Testing and Conventions

All unit tests are executed and their coverage is measured when using pytest (see the pyproject.toml file for configuration details):

python -m pip install ".[test]"
python -m pytest

Alternatively, all unit tests are included in the module itself and can be executed using doctest:

python src/egcd/egcd.py -v

Style conventions are enforced using Pylint:

python -m pip install ".[lint]"
python -m pylint src/egcd

Contributions

In order to contribute to the source code, open an issue or submit a pull request on the GitHub page for this library.

Versioning

Beginning with version 0.1.0, the version number format for this library and the changes to the library associated with version number increments conform with Semantic Versioning 2.0.0.

Publishing

This library can be published as a package on PyPI via the GitHub Actions workflow found in .github/workflows/build-publish-sign-release.yml that follows the recommendations found in the Python Packaging User Guide.

Ensure that the correct version number appears in pyproject.toml, and that any links in this README document to the Read the Docs documentation of this package (or its dependencies) have appropriate version numbers. Also ensure that the Read the Docs project for this library has an automation rule that activates and sets as the default all tagged versions.

To publish the package, create and push a tag for the version being published (replacing ?.?.? with the version number):

git tag ?.?.?
git push origin ?.?.?