Introduction

The pyfinite package is a python package for dealing with finite fields and related mathematical operations. Also included is a generic matrix package for doing matrix operations over generic fields. As an illustration a Reed-Solomon erasure correcting code implementation is provided using these tools.

Roughly speaking a "field" is a mathematical space where consistent addition, subtraction, multiplication, and division operations are defined. A "finite field" is a field where the number of elements is finite. Perhaps the most familiar finite field is the Boolean field where the elements are 0 and 1, addition (and subtraction) correspond to XOR, and multiplication (and division) work as normal for 0 and 1.

More complicated finite fields are useful and interesting for cryptography and erasure correcting codes.

Usage

After you install via something like pip install pyfinite, the best way to get started is to look at the doctest examples in the following files:

• ffield.py: See docstring for FField and FElement classes.
  • This shows you how to work with finite fields.
• genericmatrix.py: See docstring for GenericMatrix class.
  • This shows you how to do matrix operations on a generic field.
• rs_code.py: See docstring for RSCode class.
  • This shows you how to do Reed-Solomon erasure correcting codes.
• file_ecc.py: See the top-level docstring for the file_ecc module.
  • Shows you how to encode a file into multiple pieces and decode from a subset of those pieces.

For example, after you install pyfinite and start the python interpreter, do something like the following to see help on finite fields:

>>> from pyfinite import ffield
>>> help(ffield.FField)

or if you want to dive right in, you can try something like the following:

>>> from pyfinite import ffield
>>> F = ffield.FField(5) # create the field GF(2^5)
>>> a = 7    # field elements are denoted as integers from 0 to 2^5-1
>>> b = 15
>>> F.ShowPolynomial(a) # show the polynomial representation of a
'x^2 + x^1 + 1'
>>> c = F.Multiply(a,b) # multiply a and b modulo the field generator
>>> c
8
>>> F.ShowPolynomial(c)
'x^3'

Alternatively, you can jump into the genericmatrix.py package with something like:

>>> import genericmatrix
>>> v = genericmatrix.GenericMatrix((3,3))
>>> v.SetRow(0,[0.0, -1.0, 1.0])
>>> v.SetRow(1,[1.0, 1.0, 1.0])
>>> v.SetRow(2,[1.0, 1.0, -1.0])
>>> v
<matrix
  0.0 -1.0  1.0
  1.0  1.0  1.0
  1.0  1.0 -1.0>
>>> vi = v.Inverse()

Then for some real fun, you can try experimenting with generic matrix operations on elements of a finite field! The nice thing about the genericmatrix module is that it only relies on the standard python arithmetic operators so you can use it for anything with sane +, -, *, and / operators. See the help on genericmatrix for more info.

Finally, if you just want erasure correction, see the docs for the rs_code and file_ecc modules via something like

>>> import rs_code, file_ecc
>>> help(file_ecc)
>>> help(rs_code)

Future work

This code was written many years ago and hosted on an old MIT web site under the name py_ecc before being moved to github. It is in need of some love. In particular, it could use:

1. Reworking to fix pep8/pylint warnings and generally better python style.
2. More documentation.
3. More examples.
4. Travis setup to verify doctests in both python2 and python3.
   • These have been manually verified but it would be nice to have a setup which can run tests on multiple versions of python in an automated way.

To help or contribute please see the main project site at https://github.com/emin63/pyfinite.