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: 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 \(m\).

See also

is_primitive_element, FieldArray.primitive_elements

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

Construct the extension field \(\mathrm{GF}(3^4)\).

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

In [2]: GF = galois.GF(3**4, irreducible_poly=f, repr="poly")

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

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

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

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

In [5]: phi = galois.euler_phi(3**4 - 1); phi
Out[5]: 16

In [6]: len(g) == phi
Out[6]: False

Shows that each primitive element has an order of \(q^m - 1\).

# Convert the polynomials over GF(3) into elements of GF(3^4)
In [7]: g = GF([int(gi) for gi in g]); g
Out[7]: 
GF([                  α,               α + 1,                  2α,
                 2α + 2,             α^2 + 1,        α^2 + 2α + 2,
               2α^2 + 2,        2α^2 + α + 1,                 α^3,
                α^3 + 1,           α^3 + α^2,       α^3 + α^2 + 2,
          α^3 + α^2 + α,      α^3 + α^2 + 2α,  α^3 + α^2 + 2α + 2,
             α^3 + 2α^2,      α^3 + 2α^2 + 2,      α^3 + 2α^2 + α,
     α^3 + 2α^2 + α + 1, α^3 + 2α^2 + 2α + 1,                2α^3,
               2α^3 + 2,          2α^3 + α^2,      2α^3 + α^2 + 1,
     2α^3 + α^2 + α + 2,     2α^3 + α^2 + 2α, 2α^3 + α^2 + 2α + 2,
            2α^3 + 2α^2,     2α^3 + 2α^2 + 1,     2α^3 + 2α^2 + α,
    2α^3 + 2α^2 + α + 1,    2α^3 + 2α^2 + 2α], order=3^4)

In [8]: np.all(g.multiplicative_order() == GF.order - 1)
Out[8]: np.True_