-
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([194, 8, 143, 164, 105, 105, 202, 10, 72, 90], order=3^5) In [4]: i = x.log(); i Out[4]: array([141, 190, 26, 68, 150, 150, 82, 46, 192, 48]) 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([ 2α^2 + α + 2, α^4 + 2α^3 + 2α^2 + 2α + 2, α + 2, 2α^4 + α^3 + α, 2α^2 + α, 2α^3 + α^2 + 2α + 2, 2α^4 + α^3 + 2α^2 + α + 1, α^4 + α^3 + α^2, α^4 + α + 2, α^3 + α^2 + α], order=3^5) In [9]: i = x.log(); i Out[9]: array([ 17, 203, 5, 158, 127, 182, 178, 12, 73, 11]) 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^68, α^26, α^74, α^151, α^138, α^207, α^15, α^47, α^66, α^153], order=3^5) In [14]: i = x.log(); i Out[14]: array([ 68, 26, 74, 151, 138, 207, 15, 47, 66, 153]) 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([ 54, 42, 194, 95, 74, 111, 229, 169, 88, 61]) 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([ 54, 42, 194, 95, 74, 111, 229, 169, 88, 61]) 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([ 54, 42, 194, 95, 74, 111, 229, 169, 88, 61]) 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(92, order=3^5) In [27]: bases = GF.primitive_elements In [28]: i = x.log(bases); i Out[28]: array([ 14, 144, 148, 128, 10, 216, 204, 84, 72, 218, 74, 166, 48, 130, 232, 104, 26, 182, 82, 112, 38, 2, 68, 28, 24, 12, 40, 60, 34, 106, 50, 140, 234, 186, 46, 18, 80, 156, 116, 92, 236, 194, 208, 146, 214, 240, 226, 114, 150, 6, 178, 164, 142, 30, 62, 78, 180, 58, 210, 152, 20, 188, 52, 238, 96, 172, 200, 192, 64, 228, 16, 124, 136, 206, 118, 108, 134, 168, 86, 122, 126, 160, 162, 32, 196, 158, 56, 174, 54, 100, 230, 98, 224, 90, 8, 184, 138, 222, 170, 120, 36, 190, 202, 42, 70, 212, 102, 76, 94, 4]) In [29]: np.all(bases ** i == x) Out[29]: True
In [30]: x = GF.Random(low=1); x Out[30]: GF(α^4 + 2α^2 + α + 1, order=3^5) In [31]: bases = GF.primitive_elements In [32]: i = x.log(bases); i Out[32]: array([181, 185, 133, 151, 233, 217, 131, 239, 153, 191, 127, 141, 223, 125, 9, 221, 25, 175, 23, 117, 111, 95, 205, 241, 51, 207, 85, 67, 163, 195, 197, 237, 225, 123, 7, 129, 49, 29, 65, 135, 199, 19, 79, 159, 1, 147, 87, 91, 107, 43, 227, 167, 211, 215, 41, 75, 201, 93, 53, 81, 103, 97, 171, 173, 83, 63, 183, 45, 15, 61, 155, 203, 47, 105, 39, 169, 73, 115, 213, 229, 177, 219, 193, 189, 235, 3, 119, 37, 145, 31, 35, 57, 113, 161, 17, 149, 21, 139, 89, 13, 137, 71, 157, 59, 179, 27, 5, 101, 109, 69]) In [33]: np.all(bases ** i == x) Out[33]: True
In [34]: x = GF.Random(low=1); x Out[34]: GF(α^24, order=3^5) In [35]: bases = GF.primitive_elements In [36]: i = x.log(bases); i Out[36]: array([ 24, 74, 150, 12, 190, 232, 4, 144, 158, 28, 196, 8, 186, 50, 52, 40, 10, 70, 106, 192, 238, 38, 82, 48, 214, 228, 34, 172, 162, 78, 224, 240, 90, 146, 148, 100, 68, 60, 26, 54, 128, 56, 80, 112, 194, 204, 180, 230, 188, 114, 236, 212, 36, 86, 210, 30, 32, 134, 118, 226, 138, 184, 20, 166, 130, 122, 170, 18, 6, 218, 62, 178, 164, 42, 64, 116, 126, 46, 182, 140, 216, 136, 174, 124, 94, 98, 96, 160, 58, 206, 14, 168, 142, 16, 152, 108, 202, 104, 84, 102, 200, 222, 208, 72, 120, 156, 2, 234, 92, 76]) In [37]: np.all(bases ** i == x) Out[37]: True