galois.is_primitive_element

class galois.is_primitive_element(element, irreducible_poly)

Determines if \(g(x)\) is a primitive element of the Galois field \(\mathrm{GF}(p^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\).

The number of primitive elements of \(\mathrm{GF}(p^m)\) is \(\phi(p^m - 1)\), where \(\phi(n)\) is the Euler totient function, see galois.euler_totient.

Parameters

• element (galois.Poly) – An element \(g(x)\) of \(\mathrm{GF}(p^m)\) as a polynomial over \(\mathrm{GF}(p)\) with degree less than \(m\).

• irreducible_poly (galois.Poly) – The degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\) that defines the extension field \(\mathrm{GF}(p^m)\).

Returns

True if \(g(x)\) is a primitive element of \(\mathrm{GF}(p^m)\) with irreducible polynomial \(f(x)\).

Return type

bool

Examples

In [533]: GF = galois.GF(3)

In [534]: f = galois.Poly([1,1,2], field=GF); f
Out[534]: Poly(x^2 + x + 2, GF(3))

In [535]: galois.is_irreducible(f)
Out[535]: True

In [536]: galois.is_primitive(f)
Out[536]: True

In [537]: g = galois.Poly.Identity(GF); g
Out[537]: Poly(x, GF(3))

In [538]: galois.is_primitive_element(g, f)
Out[538]: True

In [539]: GF = galois.GF(3)

In [540]: f = galois.Poly([1,0,1], field=GF); f
Out[540]: Poly(x^2 + 1, GF(3))

In [541]: galois.is_irreducible(f)
Out[541]: True

In [542]: galois.is_primitive(f)
Out[542]: False

In [543]: g = galois.Poly.Identity(GF); g
Out[543]: Poly(x, GF(3))

In [544]: galois.is_primitive_element(g, f)
Out[544]: False