galois.FieldArray.log

galois.FieldArray.log(base: ElementLike | ArrayLike | None = None) → int | ndarray

Computes the discrete logarithm of the array \(x\) base \(\beta\).

Parameters:¶

base: ElementLike | ArrayLike | None = None¶

A primitive element or elements \(\beta\) of the finite field that is the base of the logarithm. The default is None which uses primitive_element.

Slower performance

If the FieldArray is configured to use lookup tables (ufunc_mode == "jit-lookup") and this method is invoked with a base different from primitive_element, then explicit calculation will be used (which is slower than using lookup tables).

Returns:¶

An integer array \(i\) of powers of \(\beta\) such that \(\beta^i = x\). The return array shape obeys NumPy broadcasting rules.

Examples

Compute the logarithm of \(x\) with default base \(\alpha\), which is the specified primitive element of the field.

Integer Polynomial Power

In [1]: GF = galois.GF(3**5)

In [2]: alpha = GF.primitive_element; alpha
Out[2]: GF(3, order=3^5)

In [3]: x = GF.Random(10, low=1); x
Out[3]: GF([136, 239, 224,   6, 134,  71, 109,  41,  96, 148], order=3^5)

In [4]: i = x.log(); i
Out[4]: array([ 99, 162, 155, 122,  42, 182, 165, 216,  50,  54])

In [5]: np.array_equal(alpha ** i, x)
Out[5]: True

In [6]: GF = galois.GF(3**5, repr="poly")

In [7]: alpha = GF.primitive_element; alpha
Out[7]: GF(α, order=3^5)

In [8]: x = GF.Random(10, low=1); x
Out[8]: 
GF([          α^2 + 2α + 2,           α^3 + 2α + 1,
          α^4 + α^3 + 2α^2,      2α^4 + α^3 + 2α^2,
      2α^4 + α^3 + α^2 + 2,    α^4 + 2α^2 + 2α + 2,
      α^4 + 2α^3 + α^2 + 1, 2α^4 + α^3 + 2α^2 + 2α,
             2α^3 + 2α + 2,                    α^3], order=3^5)

In [9]: i = x.log(); i
Out[9]: array([222, 198, 211,  19,  33, 105, 179, 183,  93,   3])

In [10]: np.array_equal(alpha ** i, x)
Out[10]: True

In [11]: GF = galois.GF(3**5, repr="power")

In [12]: alpha = GF.primitive_element; alpha
Out[12]: GF(α, order=3^5)

In [13]: x = GF.Random(10, low=1); x
Out[13]: 
GF([ α^79, α^123,  α^56, α^194, α^219, α^212, α^194, α^184,  α^84, α^132],
   order=3^5)

In [14]: i = x.log(); i
Out[14]: array([ 79, 123,  56, 194, 219, 212, 194, 184,  84, 132])

In [15]: np.array_equal(alpha ** i, x)
Out[15]: True

With the default argument, numpy.log() and log() are equivalent.

In [16]: np.array_equal(np.log(x), x.log())
Out[16]: True

Compute the logarithm of \(x\) with a different base \(\beta\), which is another primitive element of the field.

Integer Polynomial Power

In [17]: beta = GF.primitive_elements[-1]; beta
Out[17]: GF(242, order=3^5)

In [18]: i = x.log(beta); i
Out[18]: array([109,  87,  16,  90, 149,  26,  90,  18,  24, 176])

In [19]: np.array_equal(beta ** i, x)
Out[19]: True

In [20]: beta = GF.primitive_elements[-1]; beta
Out[20]: GF(2α^4 + 2α^3 + 2α^2 + 2α + 2, order=3^5)

In [21]: i = x.log(beta); i
Out[21]: array([109,  87,  16,  90, 149,  26,  90,  18,  24, 176])

In [22]: np.array_equal(beta ** i, x)
Out[22]: True

In [23]: beta = GF.primitive_elements[-1]; beta
Out[23]: GF(α^185, order=3^5)

In [24]: i = x.log(beta); i
Out[24]: array([109,  87,  16,  90, 149,  26,  90,  18,  24, 176])

In [25]: np.array_equal(beta ** i, x)
Out[25]: True

Compute the logarithm of a single finite field element base all of the primitive elements of the field.

Integer Polynomial Power

In [26]: x = GF.Random(low=1); x
Out[26]: GF(192, order=3^5)

In [27]: bases = GF.primitive_elements

In [28]: i = x.log(bases); i
Out[28]: 
array([158, 104,  80, 200, 182, 156, 228, 222,  52, 144,  40, 214, 196,
       188,  60, 102,  86, 118, 234,  54,  14, 230,  76,  74,  98, 170,
         2, 124,  38,  90, 184, 128,  48,  94, 208, 134,   4,  32,  30,
       174,  36,  46, 204,  92, 168,  12,  96,  42,  68, 206, 142, 226,
       116,  62, 112,  16, 130, 136, 192,  56, 122,  82, 172,  24, 150,
       178,  10,  58, 100,  84, 146, 224, 152, 216,  18,  78, 164, 202,
       210, 236, 212,   8, 238,  50,  34,  20, 148, 166, 160, 126,  72,
       138, 108, 186, 194, 106, 140, 120, 190,   6,  26,  70, 240, 232,
        64, 180, 114,  28, 162, 218])

In [29]: np.all(bases ** i == x)
Out[29]: True

In [30]: x = GF.Random(low=1); x
Out[30]: GF(2α^3 + α^2 + 2α, order=3^5)

In [31]: bases = GF.primitive_elements

In [32]: i = x.log(bases); i
Out[32]: 
array([ 18, 116,  52, 130,  82, 174, 124, 108,  58, 142,  26,   6, 200,
        98, 160,  30,  68, 234, 140, 144, 118, 210, 122,  36, 100,  50,
        86,   8, 182, 240, 168, 180, 128, 170, 232, 196, 172, 166,  80,
       222,  96,  42,  60,  84, 206,  32,  14, 112,  20, 146,  56,  38,
       148,   4, 218, 204,  24,  40,  28, 230, 164, 138, 136,  64, 158,
       152, 188,  74, 186, 224, 228, 194,   2,  92,  48, 208,  34, 216,
        76, 226, 162, 102,  70, 214,  10, 134,  72, 120, 104,  94, 192,
       126,  46,  12, 114, 202, 212,  78, 184,  16, 150, 106, 156,  54,
        90, 238,  62, 236, 190, 178])

In [33]: np.all(bases ** i == x)
Out[33]: True

In [34]: x = GF.Random(low=1); x
Out[34]: GF(α^104, order=3^5)

In [35]: bases = GF.primitive_elements

In [36]: i = x.log(bases); i
Out[36]: 
array([104, 240, 166,  52, 178, 118,  98, 140, 120, 202, 204, 196,  80,
       136,  64,  12, 124, 142,  56, 106, 144,  84, 194, 208,  40,  20,
       228, 100, 218,  96, 164,  72, 148,  68, 238,  30, 214,  18,  32,
       234, 232, 162,  24,  82,  34, 158,  54, 190,   8,  10, 216, 112,
       156,  50, 184, 130,  58,  16, 108,  92, 114, 152,   6,  74, 160,
       206, 172,  78,  26, 138, 188, 126, 146, 182, 116, 180,  62,  38,
       224,  42, 210, 186,  28, 134,   4, 102, 174,  48,  90,  86, 222,
         2, 212, 150,  94, 226, 230, 128, 122, 200,  60, 236,  14,  70,
        36, 192, 170,  46,  76, 168])

In [37]: np.all(bases ** i == x)
Out[37]: True