# 
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The `galois` library is a Python 3 package that extends NumPy arrays to operate over finite fields.
> Enjoying the library? Give us a :star: on [GitHub](https://github.com/mhostetter/galois)!
The user creates a [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass using `GF = galois.GF(p**m)`.
`GF` is a subclass of `np.ndarray` and its constructor `x = GF(array_like)` mimics the signature of `np.array()`. The
[`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) `x` is operated on like any other NumPy array except
all arithmetic is performed in $\mathrm{GF}(p^m)$, not $\mathbb{R}$.
Internally, the finite field arithmetic is implemented by replacing [NumPy ufuncs](https://numpy.org/doc/stable/reference/ufuncs.html).
The new ufuncs are written in pure Python and [just-in-time compiled](https://numba.pydata.org/numba-doc/dev/user/vectorize.html) with
[Numba](https://numba.pydata.org/). The ufuncs can be configured to use either lookup tables (for speed) or explicit calculation (for memory savings).
> **Warning**
> The algorithms implemented in the NumPy ufuncs are not constant-time, but were instead designed for performance. As such, the library could be vulnerable to a [side-channel timing attack](https://en.wikipedia.org/wiki/Timing_attack). This library is not intended for production security, but instead for research & development, reverse engineering, cryptanalysis, experimentation, and general education.
## Features
- Supports all [Galois fields](https://mhostetter.github.io/galois/latest/api/galois.GF/) $\mathrm{GF}(p^m)$, even arbitrarily-large fields!
- [**Faster**](https://mhostetter.github.io/galois/latest/performance/prime-fields/) than native NumPy! `GF(x) * GF(y)` is faster than `(x * y) % p` for $\mathrm{GF}(p)$.
- Seamless integration with NumPy -- normal NumPy functions work on [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/)s.
- Linear algebra over finite fields using normal [`np.linalg`](https://mhostetter.github.io/galois/latest/basic-usage/array-arithmetic/#linear-algebra) functions.
- Linear transforms over finite fields, such as the FFT with [`np.fft.fft()`](https://mhostetter.github.io/galois/latest/basic-usage/array-arithmetic/#advanced-arithmetic) and the NTT with [`ntt()`](https://mhostetter.github.io/galois/latest/api/galois.ntt/).
- Functions to generate [irreducible](https://mhostetter.github.io/galois/latest/api/#irreducible-polynomials), [primitive](https://mhostetter.github.io/galois/latest/api/#primitive-polynomials), and [Conway](https://mhostetter.github.io/galois/latest/api/galois.conway_poly/) polynomials.
- Univariate polynomials over finite fields with [`Poly`](https://mhostetter.github.io/galois/latest/api/galois.Poly/).
- Forward error correction codes with [`BCH`](https://mhostetter.github.io/galois/latest/api/galois.BCH/) and [`ReedSolomon`](https://mhostetter.github.io/galois/latest/api/galois.ReedSolomon/).
- Fibonacci and Galois linear-feedback shift registers over any finite field with [`FLFSR`](https://mhostetter.github.io/galois/latest/api/galois.FLFSR/) and [`GLFSR`](https://mhostetter.github.io/galois/latest/api/galois.GLFSR/).
- Various [number theoretic functions](https://mhostetter.github.io/galois/latest/api/#number-theory).
- [Integer factorization](https://mhostetter.github.io/galois/latest/api/#factorization) and accompanying algorithms.
- [Prime number generation](https://mhostetter.github.io/galois/latest/api/#prime-number-generation) and [primality testing](https://mhostetter.github.io/galois/latest/api/#primality-tests).
## Roadmap
- Elliptic curves over finite fields
- Galois ring arrays
- GPU support
## Documentation
The documentation for `galois` is located at https://mhostetter.github.io/galois/latest/.
## Getting Started
The [Getting Started](https://mhostetter.github.io/galois/latest/getting-started/) guide is intended to assist the user with installing the
library, creating two example arrays, and performing basic array arithmetic. See [Basic Usage](https://mhostetter.github.io/galois/latest/basic-usage/array-classes/)
for more detailed discussions and examples.
### Install the package
The latest version of `galois` can be installed from [PyPI](https://pypi.org/project/galois/) using `pip`.
```console
$ python3 -m pip install galois
```
Import the `galois` package in Python.
```python
In [1]: import galois
In [2]: galois.__version__
Out[2]: '0.3.8'
```
### Create a [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass
Next, create a [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass
for the specific finite field you'd like to work in. This is created using the `galois.GF()` class factory. In this example, we are
working in $\mathrm{GF}(3^5)$.
```python
In [3]: GF = galois.GF(3**5)
In [4]: print(GF.properties)
Galois Field:
name: GF(3^5)
characteristic: 3
degree: 5
order: 243
irreducible_poly: x^5 + 2x + 1
is_primitive_poly: True
primitive_element: x
```
The [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass `GF` is a subclass of
`np.ndarray` that performs all arithmetic in the Galois field $\mathrm{GF}(3^5)$, not in $\mathbb{R}$.
```python
In [5]: issubclass(GF, galois.FieldArray)
Out[5]: True
In [6]: issubclass(GF, np.ndarray)
Out[6]: True
```
See [Array Classes](https://mhostetter.github.io/galois/latest/basic-usage/array-classes/) for more details.
### Create two [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) instances
Next, create a new [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) `x` by passing an
[`ArrayLike`](https://mhostetter.github.io/galois/latest/api/galois.typing.ArrayLike/) object to `GF`'s constructor.
```python
In [7]: x = GF([236, 87, 38, 112]); x
Out[7]: GF([236, 87, 38, 112], order=3^5)
```
The array `x` is an instance of [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) and also
an instance of `np.ndarray`.
```python
In [8]: isinstance(x, galois.FieldArray)
Out[8]: True
In [9]: isinstance(x, np.ndarray)
Out[9]: True
```
Create a second [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) `y` by converting an existing
NumPy array (without copying it) by invoking `.view()`. When finished working in the finite field, view it back as a NumPy array
with `.view(np.ndarray)`.
```python
# y represents an array created elsewhere in the code
In [10]: y = np.array([109, 17, 108, 224]); y
Out[10]: array([109, 17, 108, 224])
In [11]: y = y.view(GF); y
Out[11]: GF([109, 17, 108, 224], order=3^5)
```
See [Array Creation](https://mhostetter.github.io/galois/latest/basic-usage/array-creation/) for more details.
### Change the element representation
The representation of finite field elements can be set to either the integer (`"int"`), polynomial (`"poly"`),
or power (`"power"`) representation. The default representation is the integer representation since integers are natural when
working with integer NumPy arrays.
Set the element representation by passing the `repr` keyword argument to `galois.GF()` or by calling the `repr()`
classmethod. Choose whichever element representation is most convenient.
```python
# The default is the integer representation
In [12]: x
Out[12]: GF([236, 87, 38, 112], order=3^5)
In [13]: GF.repr("poly"); x
Out[13]:
GF([2α^4 + 2α^3 + 2α^2 + 2, α^4 + 2α,
α^3 + α^2 + 2, α^4 + α^3 + α + 1], order=3^5)
In [14]: GF.repr("power"); x
Out[14]: GF([α^204, α^16, α^230, α^34], order=3^5)
# Reset to the integer representation
In [15]: GF.repr("int");
```
See [Element Representation](https://mhostetter.github.io/galois/latest/basic-usage/element-representation/) for more details.
### Perform array arithmetic
Once you have two Galois field arrays, nearly any arithmetic operation can be performed using normal NumPy arithmetic.
The traditional [NumPy broadcasting rules](https://numpy.org/doc/stable/user/basics.broadcasting.html) apply.
Standard element-wise array arithmetic -- addition, subtraction, multiplication, and division -- are easily preformed.
```python
In [16]: x + y
Out[16]: GF([ 18, 95, 146, 0], order=3^5)
In [17]: x - y
Out[17]: GF([127, 100, 173, 224], order=3^5)
In [18]: x * y
Out[18]: GF([ 21, 241, 179, 82], order=3^5)
In [19]: x / y
Out[19]: GF([ 67, 47, 192, 2], order=3^5)
```
More complicated arithmetic, like square root and logarithm base $\alpha$, are also supported.
```python
In [20]: np.sqrt(x)
Out[20]: GF([ 51, 135, 40, 16], order=3^5)
In [21]: np.log(x)
Out[21]: array([204, 16, 230, 34])
```
See [Array Arithmetic](https://mhostetter.github.io/galois/latest/basic-usage/array-arithmetic/) for more details.
## Acknowledgements
The `galois` library is an extension of, and completely dependent on, [NumPy](https://numpy.org/). It also heavily
relies on [Numba](https://numba.pydata.org/) and the [LLVM just-in-time compiler](https://llvm.org/) for optimizing performance
of the finite field arithmetic.
[Frank Luebeck's compilation](http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/index.html) of Conway polynomials and
[Wolfram's compilation](https://datarepository.wolframcloud.com/resources/Primitive-Polynomials/) of primitive polynomials are used
for efficient polynomial lookup, when possible.
[The Cunningham Book's tables](https://homes.cerias.purdue.edu/~ssw/cun/third/index.html) of prime factorizations, $b^n \pm 1$
for $b \in \{2, 3, 5, 6, 7, 10, 11, 12\}$, are used to generate factorization lookup tables. These lookup tables speed-up the
creation of large finite fields by avoiding the need to factor large integers.
[Sage](https://www.sagemath.org/) is used extensively for generating test vectors for finite field arithmetic and polynomial arithmetic.
[SymPy](https://www.sympy.org/en/index.html) is used to generate some test vectors. [Octave](https://www.gnu.org/software/octave/index)
is used to generate test vectors for forward error correction codes.
This library would not be possible without all of the other libraries mentioned. Thank you to all their developers!
## Citation
If this library was useful to you in your research, please cite us. Following the [GitHub citation standards](https://docs.github.com/en/github/creating-cloning-and-archiving-repositories/creating-a-repository-on-github/about-citation-files), here is the recommended citation.
### BibTeX
```bibtex
@software{Hostetter_Galois_2020,
title = {{Galois: A performant NumPy extension for Galois fields}},
author = {Hostetter, Matt},
month = {11},
year = {2020},
url = {https://github.com/mhostetter/galois},
}
```
### APA
```
Hostetter, M. (2020). Galois: A performant NumPy extension for Galois fields [Computer software]. https://github.com/mhostetter/galois
```
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"description": "# \n\n<div align=center>\n <a href=\"https://pypi.org/project/galois\"><img src=\"https://img.shields.io/pypi/v/galois\"></a>\n <a href=\"https://pypi.org/project/galois\"><img src=\"https://img.shields.io/pypi/pyversions/galois\"></a>\n <a href=\"https://pypi.org/project/galois\"><img src=\"https://img.shields.io/pypi/wheel/galois\"></a>\n <a href=\"https://pypistats.org/packages/galois\"><img src=\"https://img.shields.io/pypi/dm/galois\"></a>\n <a href=\"https://pypi.org/project/galois\"><img src=\"https://img.shields.io/pypi/l/galois\"></a>\n <a href=\"https://twitter.com/galois_py\"><img src=\"https://img.shields.io/static/v1?label=follow&message=@galois_py&color=blue&logo=twitter\"></a>\n</div>\n\n<div align=center>\n <a href=\"https://github.com/mhostetter/galois/actions/workflows/docs.yaml\"><img src=\"https://github.com/mhostetter/galois/actions/workflows/docs.yaml/badge.svg\"></a>\n <a href=\"https://github.com/mhostetter/galois/actions/workflows/lint.yaml\"><img src=\"https://github.com/mhostetter/galois/actions/workflows/lint.yaml/badge.svg\"></a>\n <a href=\"https://github.com/mhostetter/galois/actions/workflows/build.yaml\"><img src=\"https://github.com/mhostetter/galois/actions/workflows/build.yaml/badge.svg\"></a>\n <a href=\"https://github.com/mhostetter/galois/actions/workflows/test.yaml\"><img src=\"https://github.com/mhostetter/galois/actions/workflows/test.yaml/badge.svg\"></a>\n <a href=\"https://codecov.io/gh/mhostetter/galois\"><img src=\"https://codecov.io/gh/mhostetter/galois/branch/main/graph/badge.svg?token=3FJML79ZUK\"></a>\n</div>\n\nThe `galois` library is a Python 3 package that extends NumPy arrays to operate over finite fields.\n\n> Enjoying the library? Give us a :star: on [GitHub](https://github.com/mhostetter/galois)!\n\nThe user creates a [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass using `GF = galois.GF(p**m)`.\n`GF` is a subclass of `np.ndarray` and its constructor `x = GF(array_like)` mimics the signature of `np.array()`. The\n[`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) `x` is operated on like any other NumPy array except\nall arithmetic is performed in $\\mathrm{GF}(p^m)$, not $\\mathbb{R}$.\n\nInternally, the finite field arithmetic is implemented by replacing [NumPy ufuncs](https://numpy.org/doc/stable/reference/ufuncs.html).\nThe new ufuncs are written in pure Python and [just-in-time compiled](https://numba.pydata.org/numba-doc/dev/user/vectorize.html) with\n[Numba](https://numba.pydata.org/). The ufuncs can be configured to use either lookup tables (for speed) or explicit calculation (for memory savings).\n\n> **Warning**\n> The algorithms implemented in the NumPy ufuncs are not constant-time, but were instead designed for performance. As such, the library could be vulnerable to a [side-channel timing attack](https://en.wikipedia.org/wiki/Timing_attack). This library is not intended for production security, but instead for research & development, reverse engineering, cryptanalysis, experimentation, and general education.\n\n## Features\n\n- Supports all [Galois fields](https://mhostetter.github.io/galois/latest/api/galois.GF/) $\\mathrm{GF}(p^m)$, even arbitrarily-large fields!\n- [**Faster**](https://mhostetter.github.io/galois/latest/performance/prime-fields/) than native NumPy! `GF(x) * GF(y)` is faster than `(x * y) % p` for $\\mathrm{GF}(p)$.\n- Seamless integration with NumPy -- normal NumPy functions work on [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/)s.\n- Linear algebra over finite fields using normal [`np.linalg`](https://mhostetter.github.io/galois/latest/basic-usage/array-arithmetic/#linear-algebra) functions.\n- Linear transforms over finite fields, such as the FFT with [`np.fft.fft()`](https://mhostetter.github.io/galois/latest/basic-usage/array-arithmetic/#advanced-arithmetic) and the NTT with [`ntt()`](https://mhostetter.github.io/galois/latest/api/galois.ntt/).\n- Functions to generate [irreducible](https://mhostetter.github.io/galois/latest/api/#irreducible-polynomials), [primitive](https://mhostetter.github.io/galois/latest/api/#primitive-polynomials), and [Conway](https://mhostetter.github.io/galois/latest/api/galois.conway_poly/) polynomials.\n- Univariate polynomials over finite fields with [`Poly`](https://mhostetter.github.io/galois/latest/api/galois.Poly/).\n- Forward error correction codes with [`BCH`](https://mhostetter.github.io/galois/latest/api/galois.BCH/) and [`ReedSolomon`](https://mhostetter.github.io/galois/latest/api/galois.ReedSolomon/).\n- Fibonacci and Galois linear-feedback shift registers over any finite field with [`FLFSR`](https://mhostetter.github.io/galois/latest/api/galois.FLFSR/) and [`GLFSR`](https://mhostetter.github.io/galois/latest/api/galois.GLFSR/).\n- Various [number theoretic functions](https://mhostetter.github.io/galois/latest/api/#number-theory).\n- [Integer factorization](https://mhostetter.github.io/galois/latest/api/#factorization) and accompanying algorithms.\n- [Prime number generation](https://mhostetter.github.io/galois/latest/api/#prime-number-generation) and [primality testing](https://mhostetter.github.io/galois/latest/api/#primality-tests).\n\n## Roadmap\n\n- Elliptic curves over finite fields\n- Galois ring arrays\n- GPU support\n\n## Documentation\n\nThe documentation for `galois` is located at https://mhostetter.github.io/galois/latest/.\n\n## Getting Started\n\nThe [Getting Started](https://mhostetter.github.io/galois/latest/getting-started/) guide is intended to assist the user with installing the\nlibrary, creating two example arrays, and performing basic array arithmetic. See [Basic Usage](https://mhostetter.github.io/galois/latest/basic-usage/array-classes/)\nfor more detailed discussions and examples.\n\n### Install the package\n\nThe latest version of `galois` can be installed from [PyPI](https://pypi.org/project/galois/) using `pip`.\n\n```console\n$ python3 -m pip install galois\n```\n\nImport the `galois` package in Python.\n\n```python\nIn [1]: import galois\n\nIn [2]: galois.__version__\nOut[2]: '0.3.8'\n```\n\n### Create a [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass\n\nNext, create a [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass\nfor the specific finite field you'd like to work in. This is created using the `galois.GF()` class factory. In this example, we are\nworking in $\\mathrm{GF}(3^5)$.\n\n```python\nIn [3]: GF = galois.GF(3**5)\n\nIn [4]: print(GF.properties)\nGalois Field:\n name: GF(3^5)\n characteristic: 3\n degree: 5\n order: 243\n irreducible_poly: x^5 + 2x + 1\n is_primitive_poly: True\n primitive_element: x\n```\n\nThe [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) subclass `GF` is a subclass of\n`np.ndarray` that performs all arithmetic in the Galois field $\\mathrm{GF}(3^5)$, not in $\\mathbb{R}$.\n\n```python\nIn [5]: issubclass(GF, galois.FieldArray)\nOut[5]: True\n\nIn [6]: issubclass(GF, np.ndarray)\nOut[6]: True\n```\n\nSee [Array Classes](https://mhostetter.github.io/galois/latest/basic-usage/array-classes/) for more details.\n\n### Create two [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) instances\n\nNext, create a new [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) `x` by passing an\n[`ArrayLike`](https://mhostetter.github.io/galois/latest/api/galois.typing.ArrayLike/) object to `GF`'s constructor.\n\n```python\nIn [7]: x = GF([236, 87, 38, 112]); x\nOut[7]: GF([236, 87, 38, 112], order=3^5)\n```\n\nThe array `x` is an instance of [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) and also\nan instance of `np.ndarray`.\n\n```python\nIn [8]: isinstance(x, galois.FieldArray)\nOut[8]: True\n\nIn [9]: isinstance(x, np.ndarray)\nOut[9]: True\n```\n\nCreate a second [`FieldArray`](https://mhostetter.github.io/galois/latest/api/galois.FieldArray/) `y` by converting an existing\nNumPy array (without copying it) by invoking `.view()`. When finished working in the finite field, view it back as a NumPy array\nwith `.view(np.ndarray)`.\n\n```python\n# y represents an array created elsewhere in the code\nIn [10]: y = np.array([109, 17, 108, 224]); y\nOut[10]: array([109, 17, 108, 224])\n\nIn [11]: y = y.view(GF); y\nOut[11]: GF([109, 17, 108, 224], order=3^5)\n```\n\nSee [Array Creation](https://mhostetter.github.io/galois/latest/basic-usage/array-creation/) for more details.\n\n### Change the element representation\n\nThe representation of finite field elements can be set to either the integer (`\"int\"`), polynomial (`\"poly\"`),\nor power (`\"power\"`) representation. The default representation is the integer representation since integers are natural when\nworking with integer NumPy arrays.\n\nSet the element representation by passing the `repr` keyword argument to `galois.GF()` or by calling the `repr()`\nclassmethod. Choose whichever element representation is most convenient.\n\n```python\n# The default is the integer representation\nIn [12]: x\nOut[12]: GF([236, 87, 38, 112], order=3^5)\n\nIn [13]: GF.repr(\"poly\"); x\nOut[13]:\nGF([2\u03b1^4 + 2\u03b1^3 + 2\u03b1^2 + 2, \u03b1^4 + 2\u03b1,\n \u03b1^3 + \u03b1^2 + 2, \u03b1^4 + \u03b1^3 + \u03b1 + 1], order=3^5)\n\nIn [14]: GF.repr(\"power\"); x\nOut[14]: GF([\u03b1^204, \u03b1^16, \u03b1^230, \u03b1^34], order=3^5)\n\n# Reset to the integer representation\nIn [15]: GF.repr(\"int\");\n```\n\nSee [Element Representation](https://mhostetter.github.io/galois/latest/basic-usage/element-representation/) for more details.\n\n### Perform array arithmetic\n\nOnce you have two Galois field arrays, nearly any arithmetic operation can be performed using normal NumPy arithmetic.\nThe traditional [NumPy broadcasting rules](https://numpy.org/doc/stable/user/basics.broadcasting.html) apply.\n\nStandard element-wise array arithmetic -- addition, subtraction, multiplication, and division -- are easily preformed.\n\n```python\nIn [16]: x + y\nOut[16]: GF([ 18, 95, 146, 0], order=3^5)\n\nIn [17]: x - y\nOut[17]: GF([127, 100, 173, 224], order=3^5)\n\nIn [18]: x * y\nOut[18]: GF([ 21, 241, 179, 82], order=3^5)\n\nIn [19]: x / y\nOut[19]: GF([ 67, 47, 192, 2], order=3^5)\n```\n\nMore complicated arithmetic, like square root and logarithm base $\\alpha$, are also supported.\n\n```python\nIn [20]: np.sqrt(x)\nOut[20]: GF([ 51, 135, 40, 16], order=3^5)\n\nIn [21]: np.log(x)\nOut[21]: array([204, 16, 230, 34])\n```\n\nSee [Array Arithmetic](https://mhostetter.github.io/galois/latest/basic-usage/array-arithmetic/) for more details.\n\n## Acknowledgements\n\nThe `galois` library is an extension of, and completely dependent on, [NumPy](https://numpy.org/). It also heavily\nrelies on [Numba](https://numba.pydata.org/) and the [LLVM just-in-time compiler](https://llvm.org/) for optimizing performance\nof the finite field arithmetic.\n\n[Frank Luebeck's compilation](http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/index.html) of Conway polynomials and\n[Wolfram's compilation](https://datarepository.wolframcloud.com/resources/Primitive-Polynomials/) of primitive polynomials are used\nfor efficient polynomial lookup, when possible.\n\n[The Cunningham Book's tables](https://homes.cerias.purdue.edu/~ssw/cun/third/index.html) of prime factorizations, $b^n \\pm 1$\nfor $b \\in \\{2, 3, 5, 6, 7, 10, 11, 12\\}$, are used to generate factorization lookup tables. These lookup tables speed-up the\ncreation of large finite fields by avoiding the need to factor large integers.\n\n[Sage](https://www.sagemath.org/) is used extensively for generating test vectors for finite field arithmetic and polynomial arithmetic.\n[SymPy](https://www.sympy.org/en/index.html) is used to generate some test vectors. [Octave](https://www.gnu.org/software/octave/index)\nis used to generate test vectors for forward error correction codes.\n\nThis library would not be possible without all of the other libraries mentioned. Thank you to all their developers!\n\n## Citation\n\nIf this library was useful to you in your research, please cite us. Following the [GitHub citation standards](https://docs.github.com/en/github/creating-cloning-and-archiving-repositories/creating-a-repository-on-github/about-citation-files), here is the recommended citation.\n\n### BibTeX\n\n```bibtex\n@software{Hostetter_Galois_2020,\n title = {{Galois: A performant NumPy extension for Galois fields}},\n author = {Hostetter, Matt},\n month = {11},\n year = {2020},\n url = {https://github.com/mhostetter/galois},\n}\n```\n\n### APA\n\n```\nHostetter, M. (2020). Galois: A performant NumPy extension for Galois fields [Computer software]. https://github.com/mhostetter/galois\n```\n",
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