property galois.GF2.is_primitive_poly : bool

Indicates whether the irreducible_poly is a primitive polynomial.

Notes

If the irreducible polynomial is primitive, then \(x\) is a primitive element of the finite field.

Examples

The default \(\mathrm{GF}(2^8)\) field uses a primitive polynomial.

In [1]: GF = galois.GF(2**8)

In [2]: print(GF.properties)
Galois Field:
  name: GF(2^8)
  characteristic: 2
  degree: 8
  order: 256
  irreducible_poly: x^8 + x^4 + x^3 + x^2 + 1
  is_primitive_poly: True
  primitive_element: x

In [3]: assert GF.is_primitive_poly

The \(\mathrm{GF}(2^8)\) field from AES uses a non-primitive polynomial.

In [4]: GF = galois.GF(2**8, irreducible_poly="x^8 + x^4 + x^3 + x + 1")

In [5]: print(GF.properties)
Galois Field:
  name: GF(2^8)
  characteristic: 2
  degree: 8
  order: 256
  irreducible_poly: x^8 + x^4 + x^3 + x + 1
  is_primitive_poly: False
  primitive_element: x + 1

In [6]: assert not GF.is_primitive_poly