# galois.primitive_elements¶

galois.primitive_elements(irreducible_poly: Poly) List[Poly]

Finds all primitive elements $$g$$ of the Galois field $$\mathrm{GF}(q^m)$$ with degree-$$m$$ irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$.

Parameters
irreducible_poly

The degree-$$m$$ irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$ that defines the extension field $$\mathrm{GF}(q^m)$$.

Returns

List of all primitive elements of $$\mathrm{GF}(q^m)$$ with irreducible polynomial $$f(x)$$. Each primitive element $$g$$ is a polynomial over $$\mathrm{GF}(q)$$ with degree less than $$m$$.

Notes

The number of primitive elements of $$\mathrm{GF}(q^m)$$ is $$\phi(q^m - 1)$$, where $$\phi(n)$$ is the Euler totient function. See euler_phi.

Examples

Find all primitive elements for the degree $$4$$ extension of $$\mathrm{GF}(3)$$.

In [1]: f = galois.conway_poly(3, 4); f
Out[1]: Poly(x^4 + 2x^3 + 2, GF(3))

In [2]: g = galois.primitive_elements(f); g
Out[2]:
[Poly(x, GF(3)),
Poly(x + 2, GF(3)),
Poly(2x, GF(3)),
Poly(2x + 1, GF(3)),
Poly(x^2 + x, GF(3)),
Poly(x^2 + 2x + 2, GF(3)),
Poly(2x^2 + x + 1, GF(3)),
Poly(2x^2 + 2x, GF(3)),
Poly(x^3, GF(3)),
Poly(x^3 + 2, GF(3)),
Poly(x^3 + x, GF(3)),
Poly(x^3 + x + 2, GF(3)),
Poly(x^3 + 2x + 2, GF(3)),
Poly(x^3 + x^2 + 2x, GF(3)),
Poly(x^3 + x^2 + 2x + 2, GF(3)),
Poly(x^3 + 2x^2, GF(3)),
Poly(x^3 + 2x^2 + 1, GF(3)),
Poly(x^3 + 2x^2 + x, GF(3)),
Poly(x^3 + 2x^2 + x + 1, GF(3)),
Poly(x^3 + 2x^2 + 2x + 2, GF(3)),
Poly(2x^3, GF(3)),
Poly(2x^3 + 1, GF(3)),
Poly(2x^3 + x + 1, GF(3)),
Poly(2x^3 + 2x, GF(3)),
Poly(2x^3 + 2x + 1, GF(3)),
Poly(2x^3 + x^2, GF(3)),
Poly(2x^3 + x^2 + 2, GF(3)),
Poly(2x^3 + x^2 + x + 1, GF(3)),
Poly(2x^3 + x^2 + 2x, GF(3)),
Poly(2x^3 + x^2 + 2x + 2, GF(3)),
Poly(2x^3 + 2x^2 + x, GF(3)),
Poly(2x^3 + 2x^2 + x + 1, GF(3))]


Construct the extension field $$\mathrm{GF}(3^4)$$. Note, by default, GF() uses a Conway polynomial as its irreducible polynomial.

In [3]: GF = galois.GF(3**4)

In [4]: print(GF.properties)
Galois Field:
name: GF(3^4)
characteristic: 3
degree: 4
order: 81
irreducible_poly: x^4 + 2x^3 + 2
is_primitive_poly: True
primitive_element: x

In [5]: np.array_equal([int(gi) for gi in g], GF.primitive_elements)
Out[5]: True


The number of primitive elements is given by $$\phi(q^m - 1)$$.

In [6]: phi = galois.euler_phi(3**4 - 1); phi
Out[6]: 32

In [7]: len(g) == phi
Out[7]: True


Last update: Jul 12, 2022