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([2α^4 + 2α^3 + 2α^2 + 2α + 1,                      2α + 1,
                      2α^2 + 2α,              2α^3 + α^2 + 2,
         2α^4 + 2α^3 + 2α^2 + 2,                          2α,
            2α^3 + α^2 + 2α + 2,              α^4 + 2α^2 + 1,
                      α^3 + α^2,          α^3 + 2α^2 + α + 1], order=3^5)

In [4]: i = x.log(); i
Out[4]: array([238, 126, 191, 233, 204, 122, 182,  92,  71,  61])

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([ 68,  36, 141, 153, 162, 104,  52, 130,   3, 173])

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(α^3, order=3^5)

In [11]: bases = GF.primitive_elements

In [12]: i = x.log(bases); i
Out[12]: 
array([  3, 221,  49, 183, 175,  29,  61, 139, 171, 185,  85,   1, 235,
        97,  67,   5, 213,  39, 225, 145, 181,  35, 101, 127,  57,  89,
        95, 203, 111, 161, 149, 151, 223, 109,  79,  73,  69, 189, 215,
        37, 137,   7, 131, 135, 115, 207,  83,  59, 205, 105, 211,  87,
        65,  41, 117, 155, 125,  47,  45, 119, 229,  23,  63,  51, 107,
       227, 233,  93,  31, 239, 159, 113,  81, 217, 129,  75, 167, 157,
        53, 199,  27,  17, 173, 197, 163, 103, 133, 141, 219, 177, 153,
        21, 169, 123,  19, 195, 237,  13,  71,  43,  25, 179, 147,   9,
        15, 201,  91, 241, 193, 191])

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