galois.FieldArray.additive_order

galois.FieldArray.additive_order() → int | ndarray

Computes the additive order of each element in \(x\).

Returns¶: An integer array of the additive order of each element in \(x\). The return value is a single integer if the input array \(x\) is a scalar.

Notes¶

The additive order \(a\) of \(x\) in \(\mathrm{GF}(p^m)\) is the smallest integer \(a\) such that \(x a = 0\). With the exception of 0, the additive order of every element is the finite field’s characteristic.

Examples¶

Compute the additive order of each element of \(\mathrm{GF}(3^2)\).

Integer Polynomial Power

In [1]: GF = galois.GF(3**2)

In [2]: x = GF.elements; x
Out[2]: GF([0, 1, 2, 3, 4, 5, 6, 7, 8], order=3^2)

In [3]: order = x.additive_order(); order
Out[3]: array([1, 3, 3, 3, 3, 3, 3, 3, 3])

In [4]: x * order
Out[4]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0], order=3^2)

In [5]: GF = galois.GF(3**2, repr="poly")

In [6]: x = GF.elements; x
Out[6]: 
GF([     0,      1,      2,      α,  α + 1,  α + 2,     2α, 2α + 1,
    2α + 2], order=3^2)

In [7]: order = x.additive_order(); order
Out[7]: array([1, 3, 3, 3, 3, 3, 3, 3, 3])

In [8]: x * order
Out[8]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0], order=3^2)

In [9]: GF = galois.GF(3**2, repr="power")

In [10]: x = GF.elements; x
Out[10]: GF([  0,   1, α^4,   α, α^2, α^7, α^5, α^3, α^6], order=3^2)

In [11]: order = x.additive_order(); order
Out[11]: array([1, 3, 3, 3, 3, 3, 3, 3, 3])

In [12]: x * order
Out[12]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0], order=3^2)