-
galois.FieldArray.log(base: ElementLike | ArrayLike | None =
None
) int | ndarray Computes the logarithm of the array \(x\) base \(\beta\).
Danger
If the Galois field is configured to use lookup tables,
ufunc_mode == "jit-lookup"
, and this function is invoked with a base different fromprimitive_element
, then explicit calculation will be used (which is slower than lookup tables).- 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 usesprimitive_element
.
- base: ElementLike | ArrayLike | None =
- 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) 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([ 14, 118, 81, 101, 9, 58, 167, 230, 41, 186], order=3^5) In [4]: i = x.log(); i Out[4]: array([209, 83, 4, 85, 2, 28, 9, 81, 216, 94]) 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, 2α^3 + 2α^2 + 2α, 2α^4 + 2α^3 + 1, α^3 + α^2, 2α^4 + 2α^3 + 2α^2, 2α^4 + 2α^3 + 2α^2 + α + 1, 2α^4 + α^2 + α + 1, 2α^4 + 2α^3 + 2α^2 + 2α, α^2 + 2α + 1, α^3 + α^2 + 1], order=3^5) In [9]: i = x.log(); i Out[9]: array([222, 132, 240, 71, 133, 186, 226, 237, 138, 227]) 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^171, α^61, α^4, α^81, α^100, α^133, α^65, α^66, α^93, α^169], order=3^5) In [14]: i = x.log(); i Out[14]: array([171, 61, 4, 81, 100, 133, 65, 66, 93, 169]) In [15]: np.array_equal(alpha ** i, x) Out[15]: True
With the default argument,
numpy.log()
andlog()
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.
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([239, 173, 174, 75, 236, 159, 105, 88, 113, 31]) 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([239, 173, 174, 75, 236, 159, 105, 88, 113, 31]) 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([239, 173, 174, 75, 236, 159, 105, 88, 113, 31]) 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.
In [26]: x = GF.Random(low=1); x Out[26]: GF(218, order=3^5) In [27]: bases = GF.primitive_elements In [28]: i = x.log(bases); i Out[28]: array([ 44, 176, 154, 22, 66, 22, 88, 22, 88, 132, 198, 176, 220, 132, 176, 154, 220, 88, 154, 110, 154, 110, 110, 88, 110, 176, 22, 154, 176, 22, 88, 198, 44, 66, 110, 22, 44, 110, 88, 220, 154, 22, 66, 44, 154, 132, 88, 220, 22, 88, 110, 66, 66, 198, 22, 176, 220, 44, 176, 132, 132, 176, 198, 22, 198, 22, 110, 154, 132, 198, 154, 44, 220, 198, 198, 132, 110, 44, 132, 176, 154, 88, 198, 66, 132, 220, 176, 132, 66, 176, 66, 66, 220, 110, 198, 198, 88, 110, 154, 66, 44, 44, 220, 132, 220, 44, 44, 66, 88, 220]) In [29]: np.all(bases ** i == x) Out[29]: True
In [30]: x = GF.Random(low=1); x Out[30]: GF(2α^4 + 2α^2 + α, order=3^5) In [31]: bases = GF.primitive_elements In [32]: i = x.log(bases); i Out[32]: array([171, 13, 131, 25, 53, 201, 89, 179, 67, 139, 5, 57, 85, 205, 189, 43, 41, 45, 241, 37, 153, 59, 191, 221, 103, 233, 91, 197, 35, 223, 23, 137, 127, 163, 147, 47, 61, 125, 155, 173, 65, 157, 207, 193, 21, 183, 133, 217, 69, 177, 169, 119, 75, 159, 135, 123, 107, 17, 145, 7, 227, 101, 203, 3, 49, 113, 213, 219, 73, 71, 109, 149, 19, 27, 93, 161, 81, 237, 117, 211, 87, 1, 181, 97, 95, 63, 79, 51, 141, 167, 9, 229, 195, 235, 115, 225, 199, 15, 175, 31, 215, 39, 151, 29, 129, 83, 105, 185, 111, 239]) In [33]: np.all(bases ** i == x) Out[33]: True
In [34]: x = GF.Random(low=1); x Out[34]: GF(α^52, order=3^5) In [35]: bases = GF.primitive_elements In [36]: i = x.log(bases); i Out[36]: array([ 52, 120, 204, 26, 210, 180, 170, 70, 60, 222, 102, 98, 40, 68, 32, 6, 62, 192, 28, 174, 72, 42, 218, 104, 20, 10, 114, 50, 230, 48, 82, 36, 74, 34, 240, 136, 228, 130, 16, 238, 116, 202, 12, 162, 138, 200, 148, 216, 4, 126, 108, 56, 78, 146, 92, 186, 150, 8, 54, 46, 178, 76, 124, 158, 80, 224, 86, 160, 134, 190, 94, 184, 194, 212, 58, 90, 152, 140, 112, 142, 226, 214, 14, 188, 2, 172, 208, 24, 166, 164, 232, 122, 106, 196, 168, 234, 236, 64, 182, 100, 30, 118, 128, 156, 18, 96, 206, 144, 38, 84]) In [37]: np.all(bases ** i == x) Out[37]: True