# pyAdic
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The `pyadic` library is Python 3 package that provides number types for finite fields $\mathbb{F}_p$ (`ModP`) and $p$-adic numbers $\mathbb{Q}_p$ (`PAdic`). The goal is to mimic the flexible behavior of built-in types, such as `int`, `float` and `complex`. Thus, one can mix-and-match the different number types, as long as the operations are consistent. In particular, `ModP` and `PAdic` are compatible with `fractions.Fraction`.
In addition to arithmetic operations, the pyadic library also provides the following functions:
- `rationalise` to perform rationalization ($\mathbb{F}_p\rightarrow \mathbb{Q}$ and $\mathbb{Q}_p \rightarrow \mathbb{Q}$);
- `finite_field_sqrt` and `padic_sqrt` to compute square roots (which may involve `FieldExtension`);
- `padic_log` to compute the $p$-adic logarithm.
- polynomial and rational function interpolation, see `interpolation.py` module.
A shout-out to [galois](https://github.com/mhostetter/galois) for a very nice tool. It is recommented for vectorized finite field operations, unless type compatibility is an issue. For scalar operation this repo is recommended. See performance comparison below.
## Installation
The package is available on the [Python Package Index](https://pypi.org/project/pyadic/)
```console
pip install pyadic
```
Alternativelty, it can be installed by cloning the repo
```console
git clone https://github.com/GDeLaurentis/pyadic.git path/to/repo
pip install -e path/to/repo
```
### Requirements
`pip` will automatically install the required packages, which are
```
numpy, sympy
```
Additionally, `pytest` is needed for testing.
### Testing
Extensive tests are implemented with [pytest](https://github.com/pytest-dev/pytest)
```console
pytest --cov pyadic/ --cov-report html tests/ --verbose
```
## Quick Start
```python
In [1]: from pyadic import PAdic, ModP
In [2]: from fractions import Fraction as Q
# 7/13 as a 12-digit 2147483647-adic number
In [3]: PAdic(Q(7, 13), 2147483647, 12)
Out [3]: 1817101548 + 825955248*2147483647 + 1156337348*2147483647^2 + 330382099*2147483647^3 + 1321528398*2147483647^4 + 991146298*2147483647^5 + 1817101547*2147483647^6 + 825955248*2147483647^7 + 1156337348*2147483647^8 + 330382099*2147483647^9 + 1321528398*2147483647^10 + 991146298*2147483647^11 + O(2147483647^12)
# 7/13 in F_2147483647
In [4]: ModP(Q(7, 13), 2147483647)
Out [4]: 1817101548 % 2147483647
# Mapping back to rational numbers
In [5]: from pyadic.finite_field import rationalise
In [6]: rationalise(ModP(Q(7, 13), 2147483647))
Out [6]: Fraction(7, 13)
In [7]: rationalise(PAdic(Q(7, 13), 2147483647, 12))
Out [7]: Fraction(7, 13)
```
## Perfomance comparison with [galois](https://github.com/mhostetter/galois) for finite fields
Scalar instantiation and operations are faster in pyadic
```python
import numpy
from galois import GF
from pyadic import ModP
from random import randint
GFp = GF(2 ** 31 - 1)
x = randint(0, 2 ** 31 - 1)
%timeit GFp(x)
2.84 µs ± 63.5 ns
%timeit ModP(x, 2 ** 31 - 1)
297 ns ± 0.876 ns
%timeit GFp(x) ** 2
30.1 µs ± 20.6 µs
%timeit ModP(x, 2 ** 31 - 1) ** 2
2.23 µs ± 91.8 ns
```
while galois is faster for vectorized operations (the bigger the array the bigger the gain)
```python
%timeit numpy.array([randint(0, 2 ** 31 - 1) for i in range(100)]).view(GFp) ** 2
65.6 µs ± 1.86 µs
%timeit numpy.array([ModP(randint(0, 2 ** 31 - 1), 2 ** 31 - 1) for i in range(100)]) ** 2
351 µs ± 9.28 µs
```
However, galois requires everything to be appropriately typed, while pyadic performs type-casting on-the-fly
```python
numpy.array([randint(0, 2 ** 31 - 1) for i in range(100)]).view(GFp) / 2
TypeError
numpy.array([ModP(randint(0, 2 ** 31 - 1), 2 ** 31 - 1) for i in range(100)]) / 2
array([...], dtype=object)
```
## Citation
If you found this library useful, please consider citing it
```bibtex
@inproceedings{DeLaurentis:2023qhd,
author = "De Laurentis, Giuseppe",
title = "{Lips: $p$-adic and singular phase space}",
booktitle = "{21th International Workshop on Advanced Computing and Analysis Techniques in Physics Research}: {AI meets Reality}",
eprint = "2305.14075",
archivePrefix = "arXiv",
primaryClass = "hep-th",
reportNumber = "PSI-PR-23-14",
month = "5",
year = "2023"
}
```
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"description": "# pyAdic\n\n[](https://github.com/GDeLaurentis/pyadic/actions/workflows/ci_lint.yml)\n[](https://github.com/GDeLaurentis/pyadic/actions/workflows/ci_test.yml)\n[](https://github.com/GDeLaurentis/pyadic/actions)\n[](https://gdelaurentis.github.io/pyadic/)\n[](https://pypi.org/project/pyadic/)\n[](https://pypistats.org/packages/pyadic)\n[](https://mybinder.org/v2/gh/GDeLaurentis/pyadic/HEAD)\n[](https://zenodo.org/doi/10.5281/zenodo.11114230)\n[](https://pypi.org/project/pyadic/)\n\n\nThe `pyadic` library is Python 3 package that provides number types for finite fields $\\mathbb{F}_p$ (`ModP`) and $p$-adic numbers $\\mathbb{Q}_p$ (`PAdic`). The goal is to mimic the flexible behavior of built-in types, such as `int`, `float` and `complex`. Thus, one can mix-and-match the different number types, as long as the operations are consistent. In particular, `ModP` and `PAdic` are compatible with `fractions.Fraction`.\n\nIn addition to arithmetic operations, the pyadic library also provides the following functions:\n\n- `rationalise` to perform rationalization ($\\mathbb{F}_p\\rightarrow \\mathbb{Q}$ and $\\mathbb{Q}_p \\rightarrow \\mathbb{Q}$);\n- `finite_field_sqrt` and `padic_sqrt` to compute square roots (which may involve `FieldExtension`);\n- `padic_log` to compute the $p$-adic logarithm.\n- polynomial and rational function interpolation, see `interpolation.py` module.\n\nA shout-out to [galois](https://github.com/mhostetter/galois) for a very nice tool. It is recommented for vectorized finite field operations, unless type compatibility is an issue. For scalar operation this repo is recommended. See performance comparison below.\n\n## Installation\nThe package is available on the [Python Package Index](https://pypi.org/project/pyadic/)\n```console\npip install pyadic\n```\nAlternativelty, it can be installed by cloning the repo\n```console\ngit clone https://github.com/GDeLaurentis/pyadic.git path/to/repo\npip install -e path/to/repo\n```\n\n### Requirements\n`pip` will automatically install the required packages, which are\n```\nnumpy, sympy\n```\nAdditionally, `pytest` is needed for testing.\n\n### Testing\nExtensive tests are implemented with [pytest](https://github.com/pytest-dev/pytest)\n\n```console\npytest --cov pyadic/ --cov-report html tests/ --verbose\n```\n\n## Quick Start\n\n```python\nIn [1]: from pyadic import PAdic, ModP\nIn [2]: from fractions import Fraction as Q\n\n# 7/13 as a 12-digit 2147483647-adic number\nIn [3]: PAdic(Q(7, 13), 2147483647, 12) \nOut [3]: 1817101548 + 825955248*2147483647 + 1156337348*2147483647^2 + 330382099*2147483647^3 + 1321528398*2147483647^4 + 991146298*2147483647^5 + 1817101547*2147483647^6 + 825955248*2147483647^7 + 1156337348*2147483647^8 + 330382099*2147483647^9 + 1321528398*2147483647^10 + 991146298*2147483647^11 + O(2147483647^12)\n\n# 7/13 in F_2147483647\nIn [4]: ModP(Q(7, 13), 2147483647)\nOut [4]: 1817101548 % 2147483647\n\n# Mapping back to rational numbers\nIn [5]: from pyadic.finite_field import rationalise\nIn [6]: rationalise(ModP(Q(7, 13), 2147483647))\nOut [6]: Fraction(7, 13)\nIn [7]: rationalise(PAdic(Q(7, 13), 2147483647, 12))\nOut [7]: Fraction(7, 13)\n```\n\n## Perfomance comparison with [galois](https://github.com/mhostetter/galois) for finite fields\n\nScalar instantiation and operations are faster in pyadic\n```python\nimport numpy\nfrom galois import GF\nfrom pyadic import ModP\nfrom random import randint\n\nGFp = GF(2 ** 31 - 1)\nx = randint(0, 2 ** 31 - 1)\n\n%timeit GFp(x)\n2.84 \u00b5s \u00b1 63.5 ns\n\n%timeit ModP(x, 2 ** 31 - 1)\n297 ns \u00b1 0.876 ns\n\n%timeit GFp(x) ** 2\n30.1 \u00b5s \u00b1 20.6 \u00b5s \n\n%timeit ModP(x, 2 ** 31 - 1) ** 2\n2.23 \u00b5s \u00b1 91.8 ns\n```\n\nwhile galois is faster for vectorized operations (the bigger the array the bigger the gain)\n```python\n%timeit numpy.array([randint(0, 2 ** 31 - 1) for i in range(100)]).view(GFp) ** 2\n65.6 \u00b5s \u00b1 1.86 \u00b5s \n\n%timeit numpy.array([ModP(randint(0, 2 ** 31 - 1), 2 ** 31 - 1) for i in range(100)]) ** 2\n351 \u00b5s \u00b1 9.28 \u00b5s\n```\n\nHowever, galois requires everything to be appropriately typed, while pyadic performs type-casting on-the-fly\n```python\nnumpy.array([randint(0, 2 ** 31 - 1) for i in range(100)]).view(GFp) / 2\nTypeError\n\nnumpy.array([ModP(randint(0, 2 ** 31 - 1), 2 ** 31 - 1) for i in range(100)]) / 2\narray([...], dtype=object)\n```\n\n## Citation\n\nIf you found this library useful, please consider citing it\n\n\n```bibtex\n@inproceedings{DeLaurentis:2023qhd,\n author = \"De Laurentis, Giuseppe\",\n title = \"{Lips: $p$-adic and singular phase space}\",\n booktitle = \"{21th International Workshop on Advanced Computing and Analysis Techniques in Physics Research}: {AI meets Reality}\",\n eprint = \"2305.14075\",\n archivePrefix = \"arXiv\",\n primaryClass = \"hep-th\",\n reportNumber = \"PSI-PR-23-14\",\n month = \"5\",\n year = \"2023\"\n}\n```\n\n\n",
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