unireedsolomon 1.0.6

Installation

For Python 3.7+:

pip install --upgrade unireedsolomon

For Python 2.7 and <= 3.6:

pip install --upgrade unireedsolomon==1.0.5

To install and compile the Cython speed-optimized modules cff.pyx and cpolynomial.pyx, which provide a ~4x speed boost during encoding, install Cython 0.29.x and then type:

pip install --upgrade unireedsolomon --config-setting="--build-option=--cythonize"

Quickstart

>>> import unireedsolomon as rs
>>> coder = rs.RSCoder(20,13)
>>> c = coder.encode("Hello, world!")
>>> print repr(c)
'Hello, world!\x8d\x13\xf4\xf9C\x10\xe5'
>>>
>>> r = "\0"*3 + c[3:]
>>> print repr(r)
'\x00\x00\x00lo, world!\x8d\x13\xf4\xf9C\x10\xe5'
>>>
>>> coder.decode(r)
'Hello, world!'

Description

This library implements a pure-Python documented universal Reed-Solomon error correction codec with a mathematical nomenclatura, compatible with Python 3.7+ and also with PyPy 3.

The project aims to keep a well commented and organized code with an extensive documentation and mathematical clarity of the various arithmetic operations.

How does this library differs from other Reed-Solomon libraries?

• universal: compatibility with (almost) any other Reed-Solomon codec. This means that you can choose the parameters so that you can either encode data and decode it with another RS codec, or on the opposite encode data with another RS codec and decode this data with this library.

• errors-and-erasures decoding allows to decode both erasures (where you know the position of the corrupted characters) and errors (where you don’t know where are the corrupted characters) at the same time (because you can decode more erasures than errors, so if you can provide even a few know corrupted characters positions, this can help a lot the decoder to repair the message).

• documented: following literate programming guidelines, you should understand everything you need about RS by reading the code and the comments.

• mathematical nomenclatura: for example, contrary to most other RS libraries, you will see a clear distinction between the different mathematical constructs, such as the Galois Fields numbers are clearly separated from the generic Polynomial objects, and both are separated from the Reed-Solomon algorithm, which makes use of both of those constructs. For this purpose, object-oriented programming was chosen to design the architecture of the library, although obviously at the expense of a bit of performance. However, this library favors mathematical clarity and documentation over performance (even if performance is optimized whenever possible).

• pure-Python means that there are no dependencies whatsoever apart from the Python interpreter. This means that this library should be resilient in the future (since it doesn’t depend on external libraries who can become broken with time, see software rot), and you can use it on any system where Python can be installed (including online cloud services).

The authors tried their best to extensively document the algorithms. However, a lot of the math involved is non-trivial and we can’t explain it all in the comments. To learn more about the algorithms, see these resources:

• http://en.wikipedia.org/wiki/Reed–Solomon_error_correction

• http://www.cs.duke.edu/courses/spring10/cps296.3/rs_scribe.pdf

• http://www.cs.duke.edu/courses/spring10/cps296.3/decoding_rs_scribe.pdf

• http://www.cs.cmu.edu/afs/cs.cmu.edu/project/pscico-guyb/realworld/www/reed_solomon.ps

• http://www.cs.cmu.edu/afs/cs.cmu.edu/project/pscico-guyb/realworld/www/rs_decode.ps

Also, here’s a copy of the presentation one of the authors gave to the class Spring 2010 on his experience implementing this library. The LaTeX source is in the presentation directory.

http://www.cs.duke.edu/courses/spring10/cps296.3/decoding_rs.pdf

The code was lately updated to support errors-and-erasures decoding (both at the same time), and to be universal (you can supply the parameters to be compatible with almost any other RS codec).

The codec has decent performances if you use PyPy with the fast methods (~1 MB/s), but it would be faster if we drop the object-oriented design (implementing everything in functions), but this would be at the expense of mathematical clarity. If you are interested, see the reedsolo library by Tomer Filiba, which is exactly the same implementation but only functional without objects (results in about 5x speedup).

Files

rs.py

Holds the Reed-Solomon Encoder/Decoder object

polynomial.py

Contains the Polynomial object (pure-python)

ff.py

Contains the GF2int object representing an element of the GF(2^p) field, with p being 8 by default (pure-python)

polynomial.pyx

Cython implementation of polynomial.py with equivalent functions (optional)

ff.pyx

Cython implementation of ff.py with equivalent functions (optional)

Documentation

unireedsolomon.rs.RSCoder(n, k, generator=3, prim=0x11b, fcr=1, c_exp=8)

Creates a new Reed-Solomon Encoder/Decoder object configured with the given n and k values. n is the length of a codeword, must be less than 256 k is the length of the message, must be less than n generator, prim and fcr parametrize the Galois Field that will be built c_exp is the Galois Field range (ie, 8 means GF(2^8) = GF(256)), which is both the limit for one symbol’s value, and the maximum length of a message+ecc.

The code will have error correcting power (ie, maximum number of repairable symbols) of 2*e+v <= (n-k), where e is the number of errors and v the number of erasures.

The typical RSCoder is RSCoder(255, 223)

RSCoder.encode(message, poly=False, k=None)

Encode a given string with reed-solomon encoding. Returns a byte string with the k message bytes and n-k parity bytes at the end.

If a message is < k bytes long, it is assumed to be padded at the front with null bytes (ie, a shortened Reed-Solomon code).

The sequence returned is always n bytes long.

If poly is not False, returns the encoded Polynomial object instead of the polynomial translated back to a string (useful for debugging)

You can change the length (number) of parity/ecc bytes at encoding by setting k to any value between [1, n-1]. This allows to create only one RSCoder and then use it with a variable redundancy rate.

RSCoder.encode_fast(message, poly=False, k=None)

Same as encode() but using faster algorithms and optimization tricks.

RSCoder.decode(message_ecc, nostrip=False, k=None, erasures_pos=None, only_erasures=False):

Given a received string or byte array message_ecc (composed of a message string + ecc symbols at the end), attempts to decode it. If it’s a valid codeword, or if there are no more than 2*e+v <= (n-k) erratas (called the Singleton bound), the message is returned.

You can provide the erasures positions as a list to erasures_pos. For example, if you have “hella warld” and you know that a is an erasure, you can provide the list erasures_pos=[4, 7]. You can correct twice as many erasures than errors, and if some provided erasures are wrong (they are correct symbols), then there’s no problem, they will be repaired just fine (but will count towards the Singleton bound). You can also specify that you are sure there are only erasures and no errors at all by setting only_erasures=True.

A message always has k bytes, if a message contained less it is left padded with null bytes (punctured RS code). When decoded, these leading null bytes are stripped, but that can cause problems if decoding binary data. When nostrip is True, messages returned are always k bytes long. This is useful to make sure no data is lost when decoding binary data.

Note that RS can correct errors both in the message and the ecc symbols.

RSCoder.decode_fast(message_ecc, nostrip=False, k=None, erasures_pos=None, only_erasures=False):

Same as decode() but using faster algorithms and optimization tricks.

RSCoder.check(message_ecc, k=None)

Verifies the codeword (message + ecc symbols at the end) is valid by testing that the code as a polynomial code divides g, or that the syndrome is all 0 coefficients. The result is not foolproof: if it’s False, you’re sure the message was corrupted (or that you used the wrong RS parameters), but if it’s True, it’s either that the message is correct, or that there are too many errors (ie, more than the Singleton bound) for RS to do anything about it. returns True/False

RSCoder.check_fast(message_ecc, k=None)

Same as check() but using faster algorithms and optimization tricks.

unireedsolomon.ff.find_prime_polynomials(generator=2, c_exp=8, fast_primes=False, single=False)

Compute the list of prime polynomials for the given generator and galois field characteristic exponent. You can then use this prime polynomial to specify the mandatory “prim” parameter, particularly if you are using a larger Galois Field (eg, 2^16).

Internal API

Besides the main RSCoder object, two other objects are used in this implementation: Polynomial and GF2int. Their use is not specifically tied to the coder or even to the Reed-Solomon algorithm, they are just generic mathematical constructs respectively representing polynomials and Galois field’s number of base 2.

You do not need to know about the internal API to use the RS codec, this is just left as a documentation for the reader interested into dwelling inside the mathematical constructs.

polynomial.Polynomial(coefficients=[], **sparse)

There are three ways to initialize a Polynomial object. 1) With a list, tuple, or other iterable, creates a polynomial using the items as coefficients in order of decreasing power

2) With keyword arguments such as for example x3=5, sets the coefficient of x^3 to be 5

3) With no arguments, creates an empty polynomial, equivalent to Polynomial([0])

>>> print Polynomial([5, 0, 0, 0, 0, 0])
5x^5

>>> print Polynomial(x32=5, x64=8)
8x^64 + 5x^32

>>> print Polynomial(x5=5, x9=4, x0=2)
4x^9 + 5x^5 + 2

Polynomial objects export the following standard functions that perform the expected operations using polynomial arithmetic. Arithmetic of the coefficients is determined by the type passed in, so integers or GF2int objects could be used, the Polynomial class is agnostic to the type of the coefficients.

__add__
__divmod__
__eq__
__floordiv__
__hash__
__len__
__mod__
__mul__
__ne__
__neg__
__sub__
evaluate(x)
degree()
    Returns the degree of the polynomial
get_coefficient(degree)
    Returns the coefficient of the specified term

ff.GF2int(value)

Instances of this object are elements of the field GF(2^p) and instances are integers in the range 0 to (2^p)-1. By default, the field is GF(2^8) and instances are integers in the range 0 to 255 and is defined using the irreducable polynomial 0x11b or in binary form: x^8 + x^4 + x^3 + x + 1 and using 3 as the generator for the exponent table and log table.

You can however use other parameters for the Galois Field, using the init_lut() function.

ff.find_prime_polynomials(generator=2, c_exp=8, fast_primes=False, single=False)

Find the list of prime polynomials to use to generate the look-up tables for your field.

ff.init_lut(generator=3, prim=0x11b, c_exp=8)

Generate the look-up tables given the parameters. This effectively parametrize your Galois Field (ie, generator=2, prim=0x1002d, c_exp=16) will generate a GF(2^16) field.

The GF2int class inherits from int and supports all the usual integer operations. The following methods are overridden for arithmetic in the finite field GF(2^p)

__add__
__div__
__mul__
__neg__
__pow__
__radd__
__rdiv__
__rmul__
__rsub__
__sub__
inverse()
    Multiplicative inverse in GF(2^p)

Edge cases

Although sanity checks are implemented whenever possible and when they are not too much resource consuming, there are a few cases where messages will not be decoded correctly without raising an exception:

• If an incorrect erasure location is provided, the decoding algorithm will just trust the provided locations and create a syndrome that will be wrong, resulting in an incorrect decoded message. In case reliability is critical, always use the check() method after decoding to check the decoding did not go wrong.

• Reed-Solomon algorithm is limited by the Singleton Bound, which limits not only its capacity to correct errors and erasures relatively to the number of error correction symbols, but also its ability to check if the message can be decoded or not. Indeed, if the number of errors and erasures are greater than the Singleton Bound, the decoder has no way to mathematically know for sure whether there is an error at all, it may very well be a valid message (although not the message you expect, but mathematically valid nevertheless). Hence, when the message is tampered beyond the Singleton Bound, the decoder may raise an exception, but it may also return a mathematically valid but still tampered message. Using the check() method cannot fix that either. To work around this issue, a solution is to use parity or hashing functions in parallel to the Reed-Solomon codec: use the Reed-Solomon codec to repair messages, use the parity or hashing function to check if there is any error. Due to how parity and hashing functions work, they are much less likely to produce a false negative than the Reed-Solomon algorithm. This is a general rule: error correction codes are efficient at correcting messages but not at detecting errors, hashing and parity functions are the adequate tool for this purpose.

Example implementations

Projects using unireedsolomon

• Several academic publications used unireedsolomon, see this list.

Recommended reading

• “Reed-Solomon codes for coders”, free practical beginner’s tutorial with Python code examples on WikiVersity. Partially written by one of the authors of the present software.

• “Algebraic codes for data transmission”, Blahut, Richard E., 2003, Cambridge university press. Readable online on Google Books. This book was pivotal in helping to understand the intricacies of the universal Berlekamp-Massey algorithm (see figures 7.5 and 7.10).

Authors and licence

Written from scratch by Andrew Brown <brownan@gmail.com> <brownan@cs.duke.edu> (c) 2010.

Upgraded and maintained by Stephen Karl Larroque <LRQ3000@gmail.com> in 2015-2020.

Licensed under the MIT License.