-
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([187, 54, 199, 119, 39, 22, 23, 162, 57, 18], order=3^5) In [4]: i = x.log(); i Out[4]: array([108, 124, 164, 118, 11, 101, 17, 125, 196, 123]) 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^4 + α^3 + 2, α^4 + α^3 + 2α^2 + 2, 2α^4 + 2α^3 + α^2 + 2α, 2α^3 + α + 2, α^4 + 2α^3 + 2α^2 + 2α + 1, 2α^3 + α^2 + α + 1, α^3 + α^2 + α, α^2 + 2, α^4 + 2α^3 + 2α^2 + α + 2, α^4 + 2α^3 + 2α^2], order=3^5) In [9]: i = x.log(); i Out[9]: array([119, 66, 152, 77, 184, 39, 11, 74, 145, 224]) 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^224, α^24, α^112, α^99, α^100, α^238, α^87, α^37, α^131, α^125], order=3^5) In [14]: i = x.log(); i Out[14]: array([224, 24, 112, 99, 100, 238, 87, 37, 131, 125]) 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([ 64, 76, 32, 11, 236, 68, 215, 97, 193, 53]) 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([ 64, 76, 32, 11, 236, 68, 215, 97, 193, 53]) 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([ 64, 76, 32, 11, 236, 68, 215, 97, 193, 53]) 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(166, order=3^5) In [27]: bases = GF.primitive_elements In [28]: i = x.log(bases); i Out[28]: array([ 45, 169, 9, 83, 205, 193, 189, 149, 145, 113, 65, 15, 137, 3, 37, 75, 49, 101, 229, 239, 53, 41, 63, 211, 129, 125, 215, 141, 213, 237, 57, 87, 199, 183, 217, 127, 67, 173, 79, 71, 119, 105, 29, 89, 31, 201, 35, 159, 171, 123, 19, 95, 7, 131, 61, 147, 181, 221, 191, 91, 47, 103, 219, 39, 153, 17, 107, 185, 223, 197, 207, 1, 5, 109, 241, 157, 85, 177, 69, 81, 163, 13, 175, 51, 25, 93, 59, 179, 139, 235, 117, 73, 115, 151, 43, 21, 167, 195, 97, 161, 133, 23, 27, 135, 225, 111, 155, 227, 233, 203]) In [29]: np.all(bases ** i == x) Out[29]: True
In [30]: x = GF.Random(low=1); x Out[30]: GF(α^4 + 2α^3 + 2α^2 + α, order=3^5) In [31]: bases = GF.primitive_elements In [32]: i = x.log(bases); i Out[32]: array([ 80, 166, 16, 40, 230, 128, 94, 238, 204, 174, 8, 188, 136, 86, 12, 214, 114, 72, 192, 156, 148, 46, 112, 160, 68, 34, 194, 170, 56, 18, 182, 74, 58, 164, 90, 172, 146, 200, 6, 180, 104, 106, 186, 212, 82, 196, 116, 202, 62, 138, 222, 142, 120, 206, 216, 100, 26, 124, 232, 108, 218, 210, 228, 150, 30, 84, 2, 60, 20, 162, 126, 190, 224, 140, 52, 64, 178, 234, 42, 144, 236, 50, 96, 10, 152, 4, 78, 130, 32, 122, 208, 76, 70, 134, 184, 118, 28, 24, 38, 98, 102, 14, 48, 240, 158, 36, 168, 54, 226, 92]) In [33]: np.all(bases ** i == x) Out[33]: True
In [34]: x = GF.Random(low=1); x Out[34]: GF(α^22, order=3^5) In [35]: bases = GF.primitive_elements In [36]: i = x.log(bases); i Out[36]: array([ 22, 88, 198, 132, 154, 132, 44, 132, 44, 66, 220, 88, 110, 66, 88, 198, 110, 44, 198, 176, 198, 176, 176, 44, 176, 88, 132, 198, 88, 132, 44, 220, 22, 154, 176, 132, 22, 176, 44, 110, 198, 132, 154, 22, 198, 66, 44, 110, 132, 44, 176, 154, 154, 220, 132, 88, 110, 22, 88, 66, 66, 88, 220, 132, 220, 132, 176, 198, 66, 220, 198, 22, 110, 220, 220, 66, 176, 22, 66, 88, 198, 44, 220, 154, 66, 110, 88, 66, 154, 88, 154, 154, 110, 176, 220, 220, 44, 176, 198, 154, 22, 22, 110, 66, 110, 22, 22, 154, 44, 110]) In [37]: np.all(bases ** i == x) Out[37]: True