galois.is_primitive_element¶

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 \(\mathrm{GF}(p)\).

Parameters¶

element : Poly¶

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

irreducible_poly : 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

Notes

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_phi().

Examples

In [1]: GF = galois.GF(3)

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

In [3]: galois.is_irreducible(f)
Out[3]: True

In [4]: galois.is_primitive(f)
Out[4]: True

In [5]: g = galois.Poly.Identity(GF); g
Out[5]: Poly(x, GF(3))

In [6]: galois.is_primitive_element(g, f)
Out[6]: True

In [7]: GF = galois.GF(3)

In [8]: f = galois.Poly([1,0,1], field=GF); f
Out[8]: Poly(x^2 + 1, GF(3))

In [9]: galois.is_irreducible(f)
Out[9]: True

In [10]: galois.is_primitive(f)
Out[10]: False

In [11]: g = galois.Poly.Identity(GF); g
Out[11]: Poly(x, GF(3))

In [12]: galois.is_primitive_element(g, f)
Out[12]: False