galois.primitive_elements

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

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

Parameters¶

irreducible_poly: Poly¶: The degree- $m$ irreducible polynomial $f (x)$ over $GF (q)$ that defines the extension field $GF (q^{m})$ .

Returns¶

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

Notes¶

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

Examples¶

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

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, display="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 $GF (3)$ .

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 $ϕ (q^{m} - 1)$ .

In [5]: phi = galois.euler_phi(3**4 - 1); phi
Out[5]: 32

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

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]: True