The galois
library is a Python 3 package that extends NumPy arrays to operate over finite fields.
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The user creates a 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
x
is operated on like any other NumPy array except
all arithmetic is performed in
Internally, the finite field arithmetic is implemented by replacing NumPy ufuncs. The new ufuncs are written in pure Python and just-in-time compiled with Numba. 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. This library is not intended for production security, but instead for research & development, reverse engineering, cryptanalysis, experimentation, and general education.
- Supports all Galois fields
$\mathrm{GF}(p^m)$ , even arbitrarily-large fields! -
Faster 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
s. - Linear algebra over finite fields using normal
np.linalg
functions. - Linear transforms over finite fields, such as the FFT with
np.fft.fft()
and the NTT withntt()
. - Functions to generate irreducible, primitive, and Conway polynomials.
- Univariate polynomials over finite fields with
Poly
. - Forward error correction codes with
BCH
andReedSolomon
. - Fibonacci and Galois linear-feedback shift registers over any finite field with
FLFSR
andGLFSR
. - Various number theoretic functions.
- Integer factorization and accompanying algorithms.
- Prime number generation and primality testing.
- Elliptic curves over finite fields
- Galois ring arrays
- GPU support
The documentation for galois
is located at https://mhostetter.github.io/galois/latest/.
The 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 for more detailed discussions and examples.
The latest version of galois
can be installed from PyPI using pip
.
$ python3 -m pip install galois
Import the galois
package in Python.
In [1]: import galois
In [2]: galois.__version__
Out[2]: '0.3.8'
Create a FieldArray
subclass
Next, create a 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
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
subclass GF
is a subclass of
np.ndarray
that performs all arithmetic in the Galois field
In [5]: issubclass(GF, galois.FieldArray)
Out[5]: True
In [6]: issubclass(GF, np.ndarray)
Out[6]: True
See Array Classes for more details.
Create two FieldArray
instances
Next, create a new FieldArray
x
by passing an
ArrayLike
object to GF
's constructor.
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
and also
an instance of np.ndarray
.
In [8]: isinstance(x, galois.FieldArray)
Out[8]: True
In [9]: isinstance(x, np.ndarray)
Out[9]: True
Create a second 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)
.
# 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 for more details.
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.
# 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 for more details.
Once you have two Galois field arrays, nearly any arithmetic operation can be performed using normal NumPy arithmetic. The traditional NumPy broadcasting rules apply.
Standard element-wise array arithmetic -- addition, subtraction, multiplication, and division -- are easily preformed.
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
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 for more details.
The galois
library is an extension of, and completely dependent on, NumPy. It also heavily
relies on Numba and the LLVM just-in-time compiler for optimizing performance
of the finite field arithmetic.
Frank Luebeck's compilation of Conway polynomials and Wolfram's compilation of primitive polynomials are used for efficient polynomial lookup, when possible.
The Cunningham Book's tables of prime factorizations,
Sage is used extensively for generating test vectors for finite field arithmetic and polynomial arithmetic. SymPy is used to generate some test vectors. Octave 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!
If this library was useful to you in your research, please cite us. Following the GitHub citation standards, here is the recommended citation.
@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},
}
Hostetter, M. (2020). Galois: A performant NumPy extension for Galois fields [Computer software]. https://github.com/mhostetter/galois