-
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([ 24, 87, 197, 33, 231, 185, 129, 122, 28, 143], order=3^5) In [4]: i = x.log(); i Out[4]: array([191, 16, 180, 75, 152, 38, 31, 117, 207, 26]) 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 + α^2 + α + 2, 2α^4 + α^2 + α + 2, α^4 + 2α^3 + α, 2α^2, 2α^4 + α^3 + 2α^2 + 1, α^3 + 2α^2 + 2α, α^3 + 2α^2 + α + 2, 2α^4 + 2α^3 + α^2 + 2α + 2, 2α^4 + α + 2, α^2 + 2], order=3^5) In [9]: i = x.log(); i Out[9]: array([ 63, 114, 113, 123, 98, 223, 51, 177, 9, 74]) 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^31, α^68, α^51, α^29, α^184, α^82, α^43, α^41, α^241, α^98], order=3^5) In [14]: i = x.log(); i Out[14]: array([ 31, 68, 51, 29, 184, 82, 43, 41, 241, 98]) 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([199, 54, 101, 233, 18, 58, 237, 29, 17, 28]) 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([199, 54, 101, 233, 18, 58, 237, 29, 17, 28]) 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([199, 54, 101, 233, 18, 58, 237, 29, 17, 28]) 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(149, order=3^5) In [27]: bases = GF.primitive_elements In [28]: i = x.log(bases); i Out[28]: array([ 32, 18, 200, 16, 92, 148, 86, 192, 130, 118, 100, 172, 6, 228, 150, 134, 94, 174, 222, 14, 156, 212, 190, 64, 124, 62, 126, 68, 216, 104, 218, 78, 120, 114, 36, 214, 10, 80, 196, 72, 90, 236, 26, 230, 178, 30, 240, 226, 170, 152, 234, 202, 48, 34, 38, 40, 204, 98, 238, 140, 184, 84, 188, 60, 12, 82, 146, 24, 8, 210, 2, 76, 138, 56, 166, 74, 168, 142, 162, 106, 46, 20, 232, 4, 206, 50, 128, 52, 158, 194, 180, 224, 28, 102, 122, 144, 108, 58, 112, 136, 186, 54, 116, 96, 160, 208, 164, 70, 42, 182]) In [29]: np.all(bases ** i == x) Out[29]: True
In [30]: x = GF.Random(low=1); x Out[30]: GF(2α^3 + α^2 + α + 1, order=3^5) In [31]: bases = GF.primitive_elements In [32]: i = x.log(bases); i Out[32]: array([ 39, 211, 153, 201, 97, 135, 67, 113, 45, 227, 137, 13, 151, 51, 145, 65, 107, 23, 21, 191, 175, 213, 103, 199, 15, 189, 25, 219, 233, 157, 1, 27, 237, 207, 59, 223, 171, 37, 133, 239, 87, 91, 9, 61, 43, 29, 111, 41, 3, 155, 81, 163, 119, 49, 69, 79, 173, 127, 101, 95, 73, 57, 93, 179, 181, 47, 125, 241, 161, 203, 131, 17, 85, 159, 225, 7, 235, 105, 205, 167, 109, 221, 71, 141, 183, 129, 35, 139, 185, 123, 53, 31, 19, 147, 5, 115, 177, 169, 197, 75, 83, 149, 217, 117, 195, 193, 215, 229, 89, 63]) In [33]: np.all(bases ** i == x) Out[33]: True
In [34]: x = GF.Random(low=1); x Out[34]: GF(α^89, order=3^5) In [35]: bases = GF.primitive_elements In [36]: i = x.log(bases); i Out[36]: array([ 89, 103, 163, 105, 29, 215, 35, 171, 233, 3, 21, 191, 115, 135, 213, 229, 27, 189, 141, 107, 207, 151, 173, 57, 239, 59, 237, 53, 147, 17, 145, 43, 1, 7, 85, 149, 111, 41, 167, 49, 31, 127, 95, 133, 185, 91, 123, 137, 193, 211, 129, 161, 73, 87, 83, 81, 159, 23, 125, 223, 179, 37, 175, 61, 109, 39, 217, 97, 113, 153, 119, 45, 225, 65, 197, 47, 195, 221, 201, 15, 75, 101, 131, 117, 157, 71, 235, 69, 205, 169, 183, 139, 93, 19, 241, 219, 13, 63, 9, 227, 177, 67, 5, 25, 203, 155, 199, 51, 79, 181]) In [37]: np.all(bases ** i == x) Out[37]: True