- galois.FieldArray.multiplicative_order() int | ndarray
Computes the multiplicative order \(\textrm{ord}(x)\) of each element in \(x\).
- Returns:¶
An integer array of the multiplicative order of each element in \(x\). The return value is a single integer if the input array \(x\) is a scalar.
- Raises:¶
ArithmeticError – If zero is provided as an input. The multiplicative order of 0 is not defined. There is no power of 0 that ever results in 1.
Notes¶
The multiplicative order \(\textrm{ord}(x) = a\) of \(x\) in \(\mathrm{GF}(p^m)\) is the smallest power \(a\) such that \(x^a = 1\). If \(a = p^m - 1\), \(a\) is said to be a generator of the multiplicative group \(\mathrm{GF}(p^m)^\times\).
Note,
multiplicative_order()
should not be confused withorder
. The former returns the multiplicative order ofFieldArray
elements. The latter is a property of the field, namely the finite field’s order or size.Examples¶
Compute the multiplicative order of each non-zero element of \(\mathrm{GF}(3^2)\).
In [1]: GF = galois.GF(3**2) In [2]: x = GF.units; x Out[2]: GF([1, 2, 3, 4, 5, 6, 7, 8], order=3^2) In [3]: order = x.multiplicative_order(); order Out[3]: array([1, 2, 8, 4, 8, 8, 8, 4]) In [4]: x ** order Out[4]: GF([1, 1, 1, 1, 1, 1, 1, 1], order=3^2)
In [5]: GF = galois.GF(3**2, repr="poly") In [6]: x = GF.units; x Out[6]: GF([ 1, 2, α, α + 1, α + 2, 2α, 2α + 1, 2α + 2], order=3^2) In [7]: order = x.multiplicative_order(); order Out[7]: array([1, 2, 8, 4, 8, 8, 8, 4]) In [8]: x ** order Out[8]: GF([1, 1, 1, 1, 1, 1, 1, 1], order=3^2)
In [9]: GF = galois.GF(3**2, repr="power") In [10]: x = GF.units; x Out[10]: GF([ 1, α^4, α, α^2, α^7, α^5, α^3, α^6], order=3^2) In [11]: order = x.multiplicative_order(); order Out[11]: array([1, 2, 8, 4, 8, 8, 8, 4]) In [12]: x ** order Out[12]: GF([1, 1, 1, 1, 1, 1, 1, 1], order=3^2)
The elements with \(\textrm{ord}(x) = 8\) are multiplicative generators of \(\mathrm{GF}(3^2)^\times\), which are also called primitive elements.
In [13]: GF.primitive_elements Out[13]: GF([3, 5, 6, 7], order=3^2)
In [14]: GF.primitive_elements Out[14]: GF([ α, α + 2, 2α, 2α + 1], order=3^2)
In [15]: GF.primitive_elements Out[15]: GF([ α, α^7, α^5, α^3], order=3^2)