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

Computes the logarithm of the array x base β.

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) β 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 β such that βi=x. The return array shape obeys NumPy broadcasting rules.

Examples

Compute the logarithm of x with default base α, 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([            2α^4 + α^2 + 1,          α^4 + α^3 + α + 1,
           2α^4 + 2α^2 + α + 2,        2α^4 + 2α^2 + α + 1,
                     α^4 + α^3, 2α^4 + 2α^3 + 2α^2 + α + 1,
        2α^4 + 2α^3 + 2α^2 + α,        α^4 + α^3 + α^2 + 2,
                  α^4 + 2α + 2,                α^2 + α + 2], order=3^5)

In [4]: i = x.log(); i
Out[4]: array([206,  34,  38, 156,  72, 186,  96, 118, 166, 209])

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 β, 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([128, 148,  80,  10, 228, 226,  62, 172,  82,  77])

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, order=3^5)

In [11]: bases = GF.primitive_elements

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

In [13]: np.all(bases ** i == x)
Out[13]: True