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

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

Important

If the Galois field is configured to use lookup tables, ufunc_mode == "jit-lookup", and this function is invoked with a base different from primitive_element, then explicit calculation will be used.

Parameters
base: ElementLike | ArrayLike | None = None

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

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.

In [1]: GF = galois.GF(3**5, display="poly")

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

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

In [4]: i = x.log(); i
Out[4]: array([  2,  45, 232,  94,  14, 150,  17, 130,  28,  13])

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

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

In [6]: np.array_equal(np.log(x), x.log())
Out[6]: True

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

In [7]: beta = GF.primitive_elements[-1]; beta
Out[7]: GF(2α^4 + 2α^3 + 2α^2 + 2α + 2, order=3^5)

In [8]: i = x.log(beta); i
Out[8]: array([208, 203, 170,  96,   4, 112, 195, 210,   8,  21])

In [9]: np.array_equal(beta ** i, x)
Out[9]: True

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

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

In [11]: bases = GF.primitive_elements

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

In [13]: np.all(bases ** i == x)
Out[13]: True
Last update: Jul 28, 2022