-
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([119, 61, 15, 99, 50, 147, 138, 174, 239, 156], order=3^5) In [4]: i = x.log(); i Out[4]: array([118, 170, 6, 76, 51, 62, 113, 29, 162, 80]) 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 + α^3 + 2α^2 + α + 2, α^2 + α, α^2 + 2α, α^4 + α^3 + 2α + 1, 2α^4 + 2α^3 + 1, α^3 + α^2 + 2α + 1, 2α^4 + 2α^3 + α^2 + 2α + 2, α^4 + 2α^3 + α^2 + 1, 2α^4 + α^3 + 2α^2, α^4 + α^3 + 2α^2 + α + 1], order=3^5) In [9]: i = x.log(); i Out[9]: array([ 53, 70, 6, 104, 240, 30, 177, 179, 19, 56]) 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^163, α, α^116, α^170, α^142, α^180, α^121, α^40, α^74, α^41], order=3^5) In [14]: i = x.log(); i Out[14]: array([163, 1, 116, 170, 142, 180, 121, 40, 74, 41]) 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([133, 225, 206, 14, 6, 86, 121, 46, 194, 29]) 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([133, 225, 206, 14, 6, 86, 121, 46, 194, 29]) 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([133, 225, 206, 14, 6, 86, 121, 46, 194, 29]) 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(20, order=3^5) In [27]: bases = GF.primitive_elements In [28]: i = x.log(bases); i Out[28]: array([167, 41, 227, 23, 223, 1, 169, 155, 81, 215, 53, 217, 175, 237, 19, 117, 241, 235, 183, 5, 73, 93, 137, 213, 27, 195, 45, 7, 129, 89, 147, 97, 233, 179, 203, 111, 211, 115, 191, 43, 205, 67, 113, 13, 29, 149, 103, 219, 199, 37, 49, 3, 69, 185, 221, 239, 21, 35, 85, 171, 83, 151, 119, 177, 229, 133, 225, 95, 193, 75, 139, 79, 153, 141, 163, 61, 181, 189, 127, 107, 51, 59, 31, 157, 39, 87, 63, 105, 91, 173, 47, 201, 131, 71, 9, 207, 125, 159, 161, 135, 101, 123, 197, 17, 109, 57, 145, 25, 15, 65]) In [29]: np.all(bases ** i == x) Out[29]: True
In [30]: x = GF.Random(low=1); x Out[30]: GF(α^4 + α^3 + 2α^2 + α + 1, order=3^5) In [31]: bases = GF.primitive_elements In [32]: i = x.log(bases); i Out[32]: array([ 56, 92, 108, 28, 40, 138, 90, 94, 46, 146, 54, 180, 192, 36, 202, 174, 104, 2, 86, 206, 152, 8, 30, 112, 96, 48, 160, 240, 136, 182, 200, 76, 210, 18, 184, 72, 78, 140, 222, 126, 218, 50, 106, 100, 130, 234, 178, 214, 116, 24, 228, 172, 84, 120, 6, 70, 236, 232, 114, 124, 80, 26, 208, 226, 142, 204, 74, 42, 14, 186, 64, 12, 60, 98, 230, 190, 52, 188, 102, 4, 20, 156, 164, 128, 58, 148, 224, 212, 216, 158, 194, 150, 170, 118, 32, 10, 68, 162, 196, 238, 144, 34, 82, 168, 38, 122, 166, 62, 134, 16]) In [33]: np.all(bases ** i == x) Out[33]: True
In [34]: x = GF.Random(low=1); x Out[34]: GF(α^12, order=3^5) In [35]: bases = GF.primitive_elements In [36]: i = x.log(bases); i Out[36]: array([ 12, 158, 196, 6, 216, 116, 2, 72, 200, 14, 98, 4, 214, 146, 26, 20, 126, 156, 174, 96, 240, 140, 162, 24, 228, 114, 138, 86, 202, 160, 112, 120, 166, 194, 74, 50, 34, 30, 134, 148, 64, 28, 40, 56, 218, 102, 90, 236, 94, 178, 118, 106, 18, 164, 226, 136, 16, 188, 180, 234, 190, 92, 10, 204, 186, 182, 206, 130, 124, 230, 152, 210, 82, 142, 32, 58, 184, 144, 212, 70, 108, 68, 208, 62, 168, 170, 48, 80, 150, 224, 128, 84, 192, 8, 76, 54, 222, 52, 42, 172, 100, 232, 104, 36, 60, 78, 122, 238, 46, 38]) In [37]: np.all(bases ** i == x) Out[37]: True