├── LICENSE
├── README.md
├── pyGF2
    ├── LICENSE
    ├── README.md
    ├── __init__.py
    ├── generic_functions.py
    ├── gf2_add.py
    ├── gf2_div.py
    ├── gf2_inv.py
    └── gf2_mul.py
└── test_pyGF2.py


/LICENSE:
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--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
 1 | # pyGF2
 2 | **Optimized Polynomial arithmetic over GF2\[x]**.
 3 | 
 4 | This module supports polynomial addition **gf2_add**, multiplication **gf2_mul**
 5 | division **gf2_div** and Extended Euclidean Algorithm **gf2_xgcd**.
 6 | 
 7 | Polynomials must be represented as ndarrays of **uint8** including zero-valued coefficients.
 8 | The rightmost array element is assumed to represent the leading coefficient.
 9 | 
10 | ----
11 | For example:
12 | 
13 |     >>>a =np.array([1, 0, 1, 1], dtype="uint8")
14 | 
15 | ----
16 | It is equivalent to the polynomial *x^3+x^2+1*.
17 | In other words the ordering is the same as the one used in MATLAB.
18 | Sympy and numpy.polys instead assume the leftmost array element to be the leading coefficient.
19 | 
20 | The use of such ordering allows to use numpy.fft to perform multiplication between polynomials.
21 | 
22 | 
23 | ----
24 | 
25 | In general a great effort has been put to speed-up all the operations making this module
26 | suitable for handling very large degree (tens of thousands) GF2\[x] polynomials such
27 | as the ones required in coding theory.
28 | 
29 | Speed-wise it is orders of magnitude faster for polynomials of large degree
30 | when it is compared to both numpy.polys functions and Sympy.galoistools.
31 | 
32 | For Example:
33 | 
34 |  - **numpy.polymul** vs **gf2_mul** for two random GF2\[x] polynomial with 100000 elements:
35 |    **5.63 s vs 105 ms**
36 | 
37 |  - **numpy.polydiv** encounters overflow problems when dealing with large polynomials.
38 |    **gf2_div** solves that by using only modulo 2 operations.
39 | 
40 |  - **sympy.galoistools.gf_gcdex** vs **gf2_xgcd** for two random GF2\[x] polynomial with 11083 elements:
41 |    **2 m 5 s vs 1.29 s**
42 |    
43 | ---
44 |  
45 | 
46 | ## Requirements : NumPy (tested with NumPy 1.13.3 and Python 3.6.3)  
47 | 
48 | ---
49 |    
50 |    
51 | 


--------------------------------------------------------------------------------
/pyGF2/LICENSE:
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--------------------------------------------------------------------------------
/pyGF2/README.md:
--------------------------------------------------------------------------------
 1 | # pyGF2
 2 | **Optimized Polynomial arithmetic over GF2\[x]**.
 3 | 
 4 | This module supports polynomial addition **gf2_add**, multiplication **gf2_mul**
 5 | division **gf2_div** and Extended Euclidean Algorithm **gf2_xgcd**.
 6 | 
 7 | Polynomials must be represented as ndarrays of **uint8** including zero-valued coefficients.
 8 | The rightmost array element is assumed to represent the leading coefficient.
 9 | 
10 | ----
11 | For example:
12 | 
13 |     >>>a =np.array([1, 0, 1, 1], dtype="uint8")
14 | 
15 | ----
16 | It is equivalent to the polynomial *x^3+x^2+1*.
17 | In other words the ordering is the same as the one used in MATLAB.
18 | Sympy and numpy.polys instead assume the leftmost array element to be the leading coefficient.
19 | 
20 | The use of such ordering allows to use numpy.fft to perform multiplication between polynomials.
21 | 
22 | 
23 | ----
24 | 
25 | In general a great effort has been put to speed-up all the operations making this module
26 | suitable for handling very large degree (tens of thousands) GF2\[x] polynomials such
27 | as the ones required in coding theory.
28 | 
29 | Speed-wise it is orders of magnitude faster for polynomials of large degree
30 | when it is compared to both numpy.polys functions and Sympy.galoistools.
31 | 
32 | For Example:
33 | 
34 |  - **numpy.polymul** vs **gf2_mul** for two random GF2\[x] polynomial with 100000 elements:
35 |    **5.63 s vs 105 ms**
36 | 
37 |  - **numpy.polydiv** encounters overflow problems when dealing with large polynomials.
38 |    **gf2_div** solves that by using only modulo 2 operations.
39 | 
40 |  - **sympy.galoistools.gf_gcdex** vs **gf2_xgcd** for two random GF2\[x] polynomial with 11083 elements:
41 |    **2 m 5 s vs 1.29 s**


--------------------------------------------------------------------------------
/pyGF2/__init__.py:
--------------------------------------------------------------------------------
1 | from pyGF2.gf2_add import gf2_add
2 | from pyGF2.gf2_mul import gf2_mul
3 | from pyGF2.gf2_div import gf2_div
4 | from pyGF2.gf2_inv import gf2_inv, gf2_xgcd
5 | from pyGF2.generic_functions import strip_zeros, zerodegree_pol, zeros, padding, to_same_dim


--------------------------------------------------------------------------------
/pyGF2/generic_functions.py:
--------------------------------------------------------------------------------
 1 | import numpy as np
 2 | 
 3 | 
 4 | def strip_zeros(a):
 5 |     """Strip un-necessary leading (rightmost) zeroes
 6 |     from a polynomial"""
 7 | 
 8 |     return np.trim_zeros(a, trim='b')
 9 | 
10 | 
11 | def check_type(a, b):
12 |     """Type check and force cast to uint8 ndarray
13 | 
14 |     Notes
15 |     -----
16 |     Ideally for best performance one should always use uint8 or bool when using this library.
17 | 
18 |     """
19 | 
20 |     if isinstance(a, np.ndarray):
21 |         a = np.array(a, dtype="uint8")
22 |     if isinstance(b, np.ndarray):
23 |         b = np.array(b, dtype="uint8")
24 | 
25 |     if a.dtype is not "uint8":
26 |         a = a.astype("uint8")
27 | 
28 |     if b.dtype is not "uint8":
29 |         b = b.astype("uint8")
30 | 
31 |     return a, b
32 | 
33 | 
34 | def padding(a, dim):
35 |     """Zero-pad input array a a to length dim, zeroes are appended at the right"""
36 | 
37 |     return np.pad(a, (0, dim-len(a)), 'constant', constant_values=(0))
38 | 
39 | 
40 | def to_same_dim(a, b):
41 |     """Given two arrays a and b returns the two arrays with the shorter zero-padded to have
42 |     the same dimension of the longer. The arrays are padded with zeroes appended to the right.
43 |     """
44 | 
45 |     if len(a) > len(b):
46 |        return a, padding(b, len(a))
47 | 
48 |     elif len(a) < len(b):
49 |         return padding(a, len(b)), b
50 | 
51 |     else:
52 |         return a, b
53 | 
54 | 
55 | def zeros(dim):
56 |     """Returns dim coefficients for -1 degree polynomial"""
57 | 
58 |     return np.zeros(dim, dtype='uint8')
59 | 
60 | 
61 | def zerodegree_pol(dim):
62 |     """Returns dim coefficients for a zero degree polynomial"""
63 | 
64 |     out = zeros(dim)
65 |     out[0] = 1
66 | 
67 |     return out
68 | 


--------------------------------------------------------------------------------
/pyGF2/gf2_add.py:
--------------------------------------------------------------------------------
 1 | import numpy as np
 2 | from pyGF2.generic_functions import strip_zeros, check_type
 3 | 
 4 | 
 5 | 
 6 | def xor(a, b):
 7 |     """Computes the element-wise XOR of two ndarrays"""
 8 | 
 9 |     return np.logical_xor(a, b, dtype='uint8').astype("uint8")
10 | 
11 | 
12 | 
13 | 
14 | def gf2_add(a, b):
15 | 
16 |     """Add two polynomials in GF(p)[x]
17 | 
18 |     Parameters
19 |     ----------
20 |     a : ndarray (uint8 or uint8) or list
21 |         Addend polynomial's coefficients.
22 |     b : ndarray (uint8 or uint8) or list
23 |         Addend polynomial's coefficients.
24 |     Returns
25 |     -------
26 |     q : ndarray of uint8
27 |         Resulting polynomial's coefficients.
28 | 
29 | 
30 |     Notes
31 |     -----
32 |     Rightmost element in the arrays is the leading coefficient of the polynomial.
33 |     In other words, the ordering for the coefficients of the polynomials is like the one used in MATLAB while
34 |     in Sympy, for example, the leftmost element is the leading coefficient.
35 | 
36 |     Examples
37 |     ========
38 | 
39 |     >>> a = np.array([1,0,1], dtype="uint8")
40 |     >>> b = np.array([1,1], dtype="uint8")
41 |     >>> gf2_add(a,b)
42 |     array([0, 1, 1], dtype=uint8)
43 | """
44 |     a, b = check_type(a, b)
45 | 
46 |     a, b = strip_zeros(a), strip_zeros(b)
47 | 
48 |     N = len(a)
49 | 
50 |     D = len(b)
51 | 
52 |     if N == D:
53 |         res = xor(a, b)
54 | 
55 |     elif N > D:
56 | 
57 |         res = np.concatenate((xor(a[:D], b), a[D:]))
58 | 
59 |     else:
60 | 
61 |         res = np.concatenate((xor(a, b[:N]), b[N:]))
62 | 
63 |     return strip_zeros(res)
64 | 
65 | 
66 | 


--------------------------------------------------------------------------------
/pyGF2/gf2_div.py:
--------------------------------------------------------------------------------
 1 | import numpy as np
 2 | from pyGF2.generic_functions import strip_zeros
 3 | 
 4 | 
 5 | def gf2_div(dividend, divisor):
 6 |     """This function implements polynomial division over GF2.
 7 | 
 8 |     Given univariate polynomials ``dividend`` and ``divisor`` with coefficients in GF2,
 9 |     returns polynomials ``q`` and ``r``
10 |     (quotient and remainder) such that ``f = q*g + r`` (operations are intended for polynomials in GF2).
11 | 
12 |     The input arrays are the coefficients (including any coefficients
13 |     equal to zero) of the dividend and "denominator
14 |     divisor polynomials, respectively.
15 |     This function was created by heavy modification of numpy.polydiv.
16 | 
17 |     Parameters
18 |     ----------
19 |     dividend : ndarray (uint8 or bool)
20 |         Dividend polynomial's coefficients.
21 |     divisor : ndarray (uint8 or bool)
22 |         Divisor polynomial's coefficients.
23 | 
24 |     Returns
25 |     -------
26 |     q : ndarray of uint8
27 |         Quotient polynomial's coefficients.
28 | 
29 |     r : ndarray of uint8
30 |         Quotient polynomial's coefficients.
31 | 
32 |     Notes
33 |     -----
34 |     Rightmost element in the arrays is the leading coefficient of the polynomial.
35 |     In other words, the ordering for the coefficients of the polynomials is like the one used in MATLAB while
36 |     in Sympy, for example, the leftmost element is the leading coefficient.
37 | 
38 | 
39 |     Examples
40 |     ========
41 | 
42 |     >>> x = np.array([1, 0, 1, 1, 1, 0, 1], dtype="uint8")
43 |     >>> y = np.array([1, 1, 1], dtype="uint8")
44 |     >>> gf2_div(x, y)
45 |     (array([1, 1, 1, 1, 1], dtype=uint8), array([], dtype=uint8))
46 | 
47 |     """
48 | 
49 |     N = len(dividend) - 1
50 |     D = len(divisor) - 1
51 | 
52 |     if dividend[N] == 0 or divisor[D] == 0:
53 |         dividend, divisor = strip_zeros(dividend), strip_zeros(divisor)
54 | 
55 |     if not divisor.any():  # if every element is zero
56 |         raise ZeroDivisionError("polynomial division")
57 |     elif D > N:
58 |         q = np.array([])
59 |         return q, dividend
60 | 
61 |     else:
62 |         u = dividend.astype("uint8")
63 |         v = divisor.astype("uint8")
64 | 
65 |         m = len(u) - 1
66 |         n = len(v) - 1
67 |         scale = v[n].astype("uint8")
68 |         q = np.zeros((max(m - n + 1, 1),), u.dtype)
69 |         r = u.astype(u.dtype)
70 | 
71 |         for k in range(0, m - n + 1):
72 |             d = scale and r[m - k].astype("uint8")
73 |             q[-1 - k] = d
74 |             r[m - k - n:m - k + 1] = np.logical_xor(r[m - k - n:m - k + 1], np.logical_and(d, v))
75 | 
76 |         r = strip_zeros(r)
77 | 
78 |     return q, r
79 | 


--------------------------------------------------------------------------------
/pyGF2/gf2_inv.py:
--------------------------------------------------------------------------------
  1 | import numpy as np
  2 | from pyGF2.generic_functions import strip_zeros
  3 | from pyGF2.gf2_div import gf2_div
  4 | from pyGF2.gf2_add import gf2_add
  5 | 
  6 | 
  7 | def gf2_inv(f, g):
  8 |     """ Given a polynomial ``f`` and an irriducible polynomial ``g`` both in GF(p)[x], computes the
  9 |         multiplicative inverse ``out``, such that f*out == 1 mod(g) (All operations are intended in GF(p)[x]).
 10 | 
 11 |         Parameters
 12 |         ----------
 13 |         f : ndarray (uint8 or bool) or list
 14 |             Input polynomial's coefficients.
 15 |         g : ndarray (uint8 or bool) or list
 16 |             Irriducible polynomial's coefficients.
 17 | 
 18 |         Returns
 19 |         -------
 20 |         out : ndarray of uint8
 21 |             Multiplicative inverse polynomial's coefficients.
 22 | 
 23 |         Notes
 24 |         -----
 25 |         Rightmost element in the arrays is the leading coefficient of the polynomial.
 26 |         In other words, the ordering for the coefficients of the polynomials is like the one used in MATLAB while
 27 |         in Sympy, for example, the leftmost element is the leading coefficient.
 28 | 
 29 | 
 30 |         Examples
 31 |         ========
 32 | 
 33 |         >>> x = np.array([1, 1, 0, 1], dtype="uint8")
 34 |         >>> y = np.array([1, 0, 0, 0, 0, 1], dtype="uint8")
 35 |         >>> gf2_inv(x,y)
 36 |         array([0, 1, 1, 1], dtype=uint8)
 37 | 
 38 |         """
 39 | 
 40 |     out = gf2_xgcd(f, g)[0]
 41 | 
 42 |     return out
 43 | 
 44 | 
 45 | def gf2_xgcd(b, a):
 46 |     """Perform Extended Euclidean Algorithm over GF2
 47 | 
 48 |     Given polynomials ``b`` and ``a`` in ``GF(p)[x]``, computes polynomials
 49 |     ``s``, ``t`` and ``h``, such that ``h = gcd(f, g)`` and ``s*b + t*a = h``.
 50 |     The typical application of EEA is solving polynomial diophantine equations and findining multiplicative inverse.
 51 | 
 52 | 
 53 |     Parameters
 54 |     ----------
 55 |     b : ndarray (uint8 or bool) or list
 56 |         b polynomial's coefficients.
 57 |     a : ndarray (uint8 or bool) or list
 58 |         a polynomial's coefficients.
 59 |     Returns
 60 |     -------
 61 |     y2 : ndarray of uint8
 62 |          s polynomial's coefficients.
 63 |     x2 : ndarray of uint8
 64 |          t polynomial's coefficients.
 65 |     b : ndarray of uint8
 66 |         h polynomial's coefficients.
 67 | 
 68 |     Notes
 69 |     -----
 70 |     Rightmost element in the arrays is the leading coefficient of the polynomial.
 71 |     In other words, the ordering for the coefficients of the polynomials is like the one used in MATLAB while
 72 |     in Sympy, for example, the leftmost element is the leading coefficient.
 73 | 
 74 |     Examples
 75 |     ========
 76 | 
 77 |     >>> x = np.array([1, 1, 1, 1, 1, 0, 1, 0, 1], dtype="uint8")
 78 |     >>> y = np.array([1, 0, 1], dtype="uint8")
 79 |     >>> gf2_xgcd(x,y)
 80 |     (array([0, 1, 1, 1], dtype=uint8),
 81 |      array([1, 1], dtype=uint8),
 82 |      array([1], dtype=uint8))
 83 | 
 84 |     """
 85 | 
 86 |     x1 = np.array([1], dtype="uint8")
 87 |     y0 = np.array([1], dtype="uint8")
 88 | 
 89 |     x0 = np.array([], dtype="uint8")
 90 |     y1 = np.array([], dtype="uint8")
 91 | 
 92 |     while True:
 93 | 
 94 |         q, r = gf2_div(b, a)
 95 | 
 96 |         b = a
 97 | 
 98 |         if not r.any():
 99 |             break
100 | 
101 |         a = r
102 | 
103 |         if not (q.any() and x1.any()):  # if q is zero or x1 is zero
104 |             x2 = x0
105 |         elif not x0.any():  # if x0 is zero
106 |             x2 = mul(x1, q)
107 |         else:
108 |             mulres = mul(x1, q)
109 | 
110 |             x2 = gf2_add(x0, mulres)
111 | 
112 |         if not (q.any() and y1.any()):
113 |             y2 = y0
114 |         elif not y0.any():
115 |             y2 = mul(y1, q)
116 |         else:
117 |             mulres = mul(y1, q)
118 | 
119 |             y2 = gf2_add(y0, mulres)
120 | 
121 |         # update
122 |         y0 = y1
123 |         x0 = x1
124 |         y1 = y2
125 |         x1 = x2
126 | 
127 |     return y2, x2, b
128 | 
129 | 
130 | def mul(a, b):
131 |     """Performs polynomial multiplication over GF2.
132 | 
133 |        Parameters
134 |        ----------
135 |        b : ndarray (uint8 or bool) or list
136 |            Multiplicand polynomial's coefficients.
137 |        a : ndarray (uint8 or bool) or list
138 |            Multiplier polynomial's coefficients.
139 |        Returns
140 |        -------
141 |        out : ndarray of uint8
142 | 
143 | 
144 |        Notes
145 |        -----
146 |        This function performs exactly the same operation as gf2_mul but here instead of the fft, convolution
147 |        in time domain is used. This is because this function must be used multiple times in gf2_xgcd and performing the
148 |        fft in that instance introduced significant overhead.
149 |     """
150 | 
151 |     out = np.mod(np.convolve(a, b), 2).astype("uint8")
152 | 
153 |     return strip_zeros(out)
154 | 


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/pyGF2/gf2_mul.py:
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 1 | import numpy as np
 2 | from pyGF2.generic_functions import strip_zeros
 3 | 
 4 | def gf2_mul(a, b):
 5 |     """Multiply polynomials in GF(2), FFT instead of convolution in time domain is used
 6 |        to speed up computation significantly.
 7 | 
 8 |     Parameters
 9 |     ----------
10 |     a : ndarray (uint8 or bool) or list
11 |         Multiplicand polynomial's coefficients.
12 |     b : ndarray (uint8 or bool) or list
13 |         Multiplier polynomial's coefficients.
14 |     Returns
15 |     -------
16 |     q : ndarray of uint8
17 |         Resulting polynomial's coefficients.
18 | 
19 |     Examples
20 |     ========
21 | 
22 |     >>> a = np.array([1,0,1], dtype="uint8")
23 |     >>> b = np.array([1,1,1], dtype="uint8")
24 |     >>> gf2_mul(a,b)
25 |     array([1, 1, 0, 1, 1], dtype=uint8)
26 | """
27 | 
28 |     fsize = len(a) + len(b) - 1
29 | 
30 |     fsize = 2**np.ceil(np.log2(fsize)).astype(int) #use nearest power of two much faster
31 | 
32 |     fslice = slice(0, fsize)
33 | 
34 |     ta = np.fft.fft(a, fsize)
35 |     tb = np.fft.fft(b, fsize)
36 | 
37 |     res = np.fft.ifft(ta*tb)[fslice].copy()
38 | 
39 |     k = np.mod(np.rint(np.real(res)), 2).astype('uint8')
40 | 
41 |     return strip_zeros(k)


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/test_pyGF2.py:
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  1 | import unittest
  2 | import numpy as np
  3 | from pyGF2 import gf2_add, gf2_mul, gf2_div, gf2_xgcd, strip_zeros
  4 | 
  5 | 
  6 | class test_gf2(unittest.TestCase):
  7 |     """Test pyGF2 functions with random polynomials"""
  8 | 
  9 | 
 10 | 
 11 |     def test_gf2_add(self):
 12 |         """Check if addition and subtraction are the same"""
 13 | 
 14 | 
 15 |         for i in range(100):
 16 |             degree1 = np.random.randint(0, 11084, 1)
 17 |             degree2 = np.random.randint(0, 11084, 1)
 18 | 
 19 |             a = np.random.randint(0, 2, degree1, dtype='uint8')
 20 |             b = np.random.randint(0, 2, degree2, dtype='uint8')
 21 | 
 22 |             sum = gf2_add(a, b)
 23 | 
 24 |             a_2 = gf2_add(sum, b)
 25 |             b_2 = gf2_add(sum, a)
 26 | 
 27 |             assert np.array_equal(a_2, strip_zeros(a)) == True
 28 |             assert np.array_equal(b_2, strip_zeros(b)) == True
 29 | 
 30 | 
 31 |     def test_gf2_mul(self):
 32 |         """Check multiplication using distributive property"""
 33 | 
 34 |         for i in range(100):
 35 | 
 36 |             degree1 = np.random.randint(0, 11084, 1)
 37 |             degree2 = np.random.randint(0, 11084, 1)
 38 |             degree3 = np.random.randint(0, 11084, 1)
 39 | 
 40 |             # distributive property
 41 | 
 42 |             a = np.random.randint(0, 2, degree1, dtype='uint8')
 43 |             b = np.random.randint(0, 2, degree2, dtype='uint8')
 44 |             c = np.random.randint(0, 2, degree3, dtype='uint8')
 45 | 
 46 |             res1 = gf2_mul(c, gf2_add(a,b))
 47 | 
 48 |             res2 = gf2_add(gf2_mul(c, a), gf2_mul(c, b))
 49 | 
 50 |             assert np.array_equal(res1, res2) == True
 51 | 
 52 |             # check if same with np.convolve
 53 | 
 54 |             res1_2  = np.mod(np.convolve(c, gf2_add(a, b)),2).astype("uint8")
 55 | 
 56 |             assert np.array_equal(res1, strip_zeros(res1_2)) == True
 57 | 
 58 | 
 59 |     def test_gf2_div(self):
 60 |         """Test polynomial divisiopn in GF2"""
 61 | 
 62 |         for i in range(100):
 63 |             
 64 |             degree1 = np.random.randint(0, 11084, 1)
 65 |             degree2 = np.random.randint(0, 11084, 1)
 66 | 
 67 |             dividend = np.random.randint(0, 2, degree1, dtype="uint8")
 68 |             divisor = np.random.randint(0, 2, degree2, dtype="uint8")
 69 | 
 70 |             quotient, remainder = gf2_div(dividend, divisor)
 71 | 
 72 |             dividend2 = gf2_add(gf2_mul(quotient, divisor), remainder)
 73 | 
 74 |             assert np.array_equal(strip_zeros(dividend), dividend2) == True
 75 | 
 76 | 
 77 |     def test_gf2_xgcd(self):
 78 |         """Test Extended Euclidean Algorithm in GF2"""
 79 | 
 80 |         p = 11083
 81 | 
 82 |         for i in range(100):
 83 | 
 84 |             a = np.random.randint(0, 2, p).astype("uint8")
 85 | 
 86 |             irr_poly = np.array([1] + [0] * (p - 1) + [1], dtype='uint8')
 87 | 
 88 |             s,t, h = gf2_xgcd(a, irr_poly)
 89 | 
 90 |             check = gf2_add( gf2_mul(a, s), gf2_mul(irr_poly, t))
 91 | 
 92 |             assert np.array_equal(check, h) == True
 93 | 
 94 | 
 95 | 
 96 | 
 97 | 
 98 | 
 99 | 
100 | 
101 | 
102 | 
103 | 
104 | 
105 | 
106 | 
107 | 
108 | 
109 | 
110 | 
111 | 
112 | 
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114 | 
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116 | 
117 | 
118 | 
119 | 


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