-
galois.FieldArray.log(base: ElementLike | ArrayLike | None =
None
) int | ndarray Computes the discrete logarithm of the array \(x\) base \(\beta\).
- Parameters:¶
- base: ElementLike | ArrayLike | None =
None
¶ A primitive element or elements \(\beta\) of the finite field that is the base of the logarithm. The default is
None
which usesprimitive_element
.Slower performance
If the
FieldArray
is configured to use lookup tables (ufunc_mode == "jit-lookup"
) and this method is invoked with a base different fromprimitive_element
, then explicit calculation will be used (which is slower than using lookup tables).
- 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([ 89, 98, 47, 63, 96, 241, 197, 25, 230, 200], order=3^5) In [4]: i = x.log(); i Out[4]: array([166, 35, 36, 128, 50, 238, 180, 88, 81, 33]) 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^4 + 2α^2 + 2α + 1, 2α^4 + α^3 + 2α^2 + 2α, α^4 + 2α^2 + 2α + 1, 2α^4 + α^3 + 2α^2 + 2α + 2, α^3 + 2α^2 + 2α, 2α^3 + α + 2, α, 2α^4 + α^3 + α + 1, α^4 + 2α^3 + α^2 + 2α, α^3 + 2], order=3^5) In [9]: i = x.log(); i Out[9]: array([108, 183, 235, 175, 223, 77, 1, 147, 52, 15]) 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^41, α^224, α^39, α^114, α^28, α^119, α^204, α^19, α^161, α^185], order=3^5) In [14]: i = x.log(); i Out[14]: array([ 41, 224, 39, 114, 28, 119, 204, 19, 161, 185]) 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([ 29, 64, 63, 240, 8, 155, 162, 161, 167, 1]) 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([ 29, 64, 63, 240, 8, 155, 162, 161, 167, 1]) 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([ 29, 64, 63, 240, 8, 155, 162, 161, 167, 1]) 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(122, order=3^5) In [27]: bases = GF.primitive_elements In [28]: i = x.log(bases); i Out[28]: array([117, 149, 217, 119, 49, 163, 201, 97, 135, 197, 169, 39, 211, 153, 193, 195, 79, 69, 63, 89, 41, 155, 67, 113, 45, 83, 75, 173, 215, 229, 3, 81, 227, 137, 177, 185, 29, 111, 157, 233, 19, 31, 27, 183, 129, 87, 91, 123, 9, 223, 1, 5, 115, 147, 207, 237, 35, 139, 61, 43, 219, 171, 37, 53, 59, 141, 133, 239, 241, 125, 151, 51, 13, 235, 191, 21, 221, 73, 131, 17, 85, 179, 213, 181, 65, 145, 105, 175, 71, 127, 159, 93, 57, 199, 15, 103, 47, 23, 107, 225, 7, 205, 167, 109, 101, 95, 161, 203, 25, 189]) In [29]: np.all(bases ** i == x) Out[29]: True
In [30]: x = GF.Random(low=1); x Out[30]: GF(2α^4 + α^3 + α^2 + α + 2, order=3^5) In [31]: bases = GF.primitive_elements In [32]: i = x.log(bases); i Out[32]: array([ 63, 43, 61, 213, 45, 125, 71, 15, 203, 13, 91, 21, 95, 101, 197, 105, 117, 93, 127, 141, 171, 9, 185, 5, 229, 175, 59, 149, 153, 235, 225, 25, 85, 111, 207, 81, 239, 97, 159, 51, 215, 147, 89, 173, 237, 233, 49, 29, 191, 27, 75, 133, 155, 135, 37, 109, 205, 19, 219, 79, 211, 241, 113, 103, 69, 169, 53, 17, 167, 179, 193, 195, 7, 201, 47, 123, 119, 151, 145, 65, 83, 115, 3, 23, 35, 227, 131, 57, 1, 87, 67, 199, 161, 163, 157, 223, 137, 31, 39, 177, 41, 129, 183, 189, 73, 107, 217, 221, 181, 139]) In [33]: np.all(bases ** i == x) Out[33]: True
In [34]: x = GF.Random(low=1); x Out[34]: GF(α^228, order=3^5) In [35]: bases = GF.primitive_elements In [36]: i = x.log(bases); i Out[36]: array([228, 98, 94, 114, 232, 26, 38, 158, 170, 24, 168, 76, 194, 112, 10, 138, 216, 60, 160, 130, 204, 240, 174, 214, 218, 230, 202, 182, 208, 136, 192, 102, 8, 56, 196, 224, 162, 86, 126, 150, 6, 48, 34, 96, 28, 2, 16, 128, 92, 236, 64, 78, 100, 212, 180, 164, 62, 184, 32, 90, 222, 54, 190, 4, 146, 70, 42, 50, 178, 14, 226, 118, 106, 36, 124, 134, 108, 74, 156, 120, 116, 82, 80, 210, 46, 84, 186, 68, 188, 142, 12, 144, 18, 152, 234, 58, 104, 20, 72, 122, 206, 52, 40, 200, 172, 30, 140, 166, 148, 238]) In [37]: np.all(bases ** i == x) Out[37]: True