Computes the multiplicative order $\text{ord}(x)$ 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 $\text{ord}(x)=a$ 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, display="poly")

In [2]: x = GF.units; x
Out[2]: 
GF([     1,      2,      α,  α + 1,  α + 2,     2α, 2α + 1, 2α + 2],
   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)

The elements with $\text{ord}(x)=8$ are multiplicative generators of $\mathrm{GF}(3^2{)}^{\times }$, which are also called primitive elements.

In [5]: GF.primitive_elements
Out[5]: GF([     α,  α + 2,     2α, 2α + 1], order=3^2)