======= shamirs

Minimal pure-Python implementation of Shamir's secret sharing scheme <https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing>__.

.. |pypi| image:: https://badge.fury.io/py/shamirs.svg# :target: https://badge.fury.io/py/shamirs :alt: PyPI version and link.

.. |readthedocs| image:: https://readthedocs.org/projects/shamirs/badge/?version=latest :target: https://shamirs.readthedocs.io/en/latest/?badge=latest :alt: Read the Docs documentation status.

.. |actions| image:: https://github.com/lapets/shamirs/workflows/lint-test-cover-docs/badge.svg# :target: https://github.com/lapets/shamirs/actions/workflows/lint-test-cover-docs.yml :alt: GitHub Actions status.

.. |coveralls| image:: https://coveralls.io/repos/github/lapets/shamirs/badge.svg?branch=main :target: https://coveralls.io/github/lapets/shamirs?branch=main :alt: Coveralls test coverage summary.

Purpose

.. |secrets_randbelow| replace:: secrets.randbelow .. _secrets_randbelow: https://docs.python.org/3/library/secrets.html#secrets.randbelow

This library provides functions and data structures for computing secret shares given an integer input value and for reassembling an integer from its corresponding secret shares via Lagrange interpolation over finite fields (according to Shamir's secret sharing scheme <https://en.wikipedia.org/wiki/Shamir%27s_secret_sharing>). The built-in |secrets_randbelow|_ function is used to generate random coefficients. The lagrange <https://pypi.org/project/lagrange> library is used for Lagrange interpolation.

Installation and Usage

This library is available as a package on PyPI <https://pypi.org/project/shamirs>__:

.. code-block:: bash

python -m pip install shamirs

The library can be imported in the usual manner:

.. code-block:: python

import shamirs

Examples ^^^^^^^^

.. |shares| replace:: shares .. _shares: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.shares

.. |interpolate| replace:: interpolate .. _interpolate: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.interpolate

The library provides the function |shares|_ for transforming a nonnegative integer plaintext into a number of secret shares and the function |interpolate|_ for reassembling those shares back into the plaintext they represent:

.. code-block:: python

>>> ss = shamirs.shares(123, quantity=3)
>>> len(ss)
3
>>> shamirs.interpolate(ss)
123
>>> ss = shamirs.shares(456, quantity=20, modulus=15485867, threshold=10)
>>> shamirs.interpolate(ss[5:15], threshold=10)
456

.. |share| replace:: share .. _share: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.share

.. |tuple| replace:: tuple .. _tuple: https://docs.python.org/3/library/stdtypes.html#tuple

Individual secret shares are represented using the |share|_ class. This class is derived from the |tuple|_ type and can have either two integer components (the share index and the share value that together determine the coordinates of a point on a polynomial curve) or three integer components (also including the modulus). One advantage of the two-component variant is that the memory footprint of |share|_ objects is reduced. These components can be accessed either directly via their indices or via named attributes:

.. code-block:: python

>>> s = shamirs.share(1, 2, 3)
>>> s.index
1
>>> s.value
2
>>> s.modulus
3
>>> [s[0], s[1], s[2]]
[1, 2, 3]
>>> int(s) # Share value.
2

The |shares|_ function accepts an optional boolean argument compact, making it possible to create secret shares that do not include the modulus component:

.. code-block:: python

>>> shamirs.shares(123, quantity=3, modulus=1009, compact=True)
[share(1, 649), share(2, 778), share(3, 510)]

.. |special_mod| replace:: __mod__ .. _special_mod: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.share.mod

It is also possible to extend a two-component |share|_ object with a modulus using the built-in modulus operator (thanks to the |special_mod|_ method):

.. code-block:: python

>>> s = shamirs.share(1, 2)
>>> s % 3
share(1, 2, 3)
>>> t = shamirs.share(1, 2)
>>> t %= 3
>>> t
share(1, 2, 3)

.. |special_add| replace:: __add__ .. _special_add: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.share.add

.. |special_mul| replace:: __mul__ .. _special_mul: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.share.mul

.. code-block:: python

>>> (r, s, t) = shamirs.shares(123, 3)
>>> (u, v, w) = shamirs.shares(456, 3)
>>> shamirs.interpolate([r + u, s + v, t + w])
579
>>> (r, s, t) = shamirs.shares(123, 3)
>>> r *= 2
>>> s *= 2
>>> t *= 2
>>> shamirs.interpolate([r, s, t])
246

When creating secret shares for a given plaintext, the modulus can be specified explicitly or omitted. When the modulus is omitted, the default is the 128-bit modulus (2 ** 127) - 1:

.. code-block:: python

>>> (r, s, t) = shamirs.shares(123, 3)
>>> r.modulus == (2 ** 127) - 1
True
>>> (r, s, t) = shamirs.shares(123, 3, modulus=1009)
>>> r.modulus
1009

When using |interpolate|_ to reconstruct a plaintext from compact |share|_ objects, the modulus must be specified explicitly:

.. code-block:: python

>>> (r, s, t) = shamirs.shares(123, 3, modulus=1009, compact=True)
>>> shamirs.interpolate([r, s, t])
Traceback (most recent call last):
  ...
ValueError: modulus is not found in share objects and is not provided as an argument
>>> shamirs.interpolate([r, s, t], modulus=1009)
123

The reconstruction threshold can also be specified explicitly or omitted. When it is omitted, the default threshold is the number of secret shares requested:

.. code-block:: python

>>> (r, s, t) = shamirs.shares(123, 3)
>>> shamirs.interpolate([r, s, t]) # Three shares (at threshold).
123
>>> shamirs.interpolate([r, s]) # Two shares (below threshold).
119174221476707020724653887077758571505
>>> (r, s, t) = shamirs.shares(123, 3, threshold=2)
>>> shamirs.interpolate([r, s]) # Two shares (at threshold).
123
>>> shamirs.interpolate([s, t]) # Two shares (at threshold).
123
>>> shamirs.interpolate([r, t]) # Two shares (at threshold).
123

The threshold argument is never required when invoking |interpolate|_, but it can reduce the number of arithmetic operations performed when reconstructing a plaintext:

.. code-block:: python

>>> ss = shamirs.shares(123, 256, threshold=2)
>>> shamirs.interpolate(ss) # Slower.
123
>>> shamirs.interpolate(ss, threshold=2) # Faster.
123

.. |add| replace:: add .. _add: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.add

.. |mul| replace:: mul .. _mul: https://shamirs.readthedocs.io/en/4.0.0/_source/shamirs.html#shamirs.shamirs.mul

To facilitate rapid prototyping and assembly of concise tests, the |add|_ and |mul|_ helper functions are provided for performing addition and scalar multiplication operations involving collections of shares:

.. code-block:: python

>>> ss = shamirs.shares(123, 3)
>>> ts = shamirs.shares(456, 3)
>>> shamirs.interpolate(shamirs.add(ss, ts))
579
>>> shamirs.interpolate(shamirs.mul(ss, 2))
246

These methods can also be used instead of built-in arithmetic operators when working with |share|_ objects that have no modulus component:

.. code-block:: python

>>> (r, s, t) = shamirs.shares(123, 3, modulus=1009, compact=True)
>>> (u, v, w) = shamirs.shares(456, 3, modulus=1009, compact=True)
>>> shamirs.interpolate(
...     [
...         shamirs.add(r, u, modulus=1009),
...         shamirs.add(s, v, modulus=1009),
...         shamirs.add(t, w, modulus=1009)
...     ],
...     modulus=1009
... )
579
>>> shamirs.interpolate(
...     [
...         shamirs.mul(r, 2, modulus=1009),
...         shamirs.mul(s, 2, modulus=1009),
...         shamirs.mul(t, 2, modulus=1009)
...     ],
...     modulus=1009
... )
246

In addition, conversion methods for bytes-like objects and Base64 strings are included to support encoding and decoding of |share|_ objects:

.. code-block:: python

>>> shamirs.share.from_base64('AQAAAAIAAADkAPED').to_bytes().hex()
'0100000002000000e400f103'
>>> [s.to_base64() for s in shamirs.shares(123, 3, 1009)]
['AQAAAAIAAADkAPED', 'AgAAAAIAAABRAfED', 'AwAAAAIAAADCAfED']

Development

All installation and development dependencies are fully specified in pyproject.toml. The project.optional-dependencies object is used to specify optional requirements <https://peps.python.org/pep-0621>__ for various development tasks. This makes it possible to specify additional options (such as docs, lint, and so on) when performing installation using pip <https://pypi.org/project/pip>__:

.. code-block:: bash

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

Documentation ^^^^^^^^^^^^^ The documentation can be generated automatically from the source files using Sphinx <https://www.sphinx-doc.org>__:

.. code-block:: bash

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

Shamirs

Install / Use

README

======= shamirs

Purpose

Installation and Usage

Development