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 with order. The former returns the multiplicative order of FieldArray 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)