- 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:¶
- 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¶
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]: True