galois.is_primitive_element¶

galois.is_primitive_element(element: PolyLike, irreducible_poly: Poly) → bool¶

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

Parameters

element: An element \(g\) of \(\mathrm{GF}(q^m)\) is a polynomial over \(\mathrm{GF}(q)\) with degree less than \(m\).
irreducible_poly: The degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\) that defines the extension field \(\mathrm{GF}(q^m)\).

Returns

True if \(g\) is a primitive element of \(\mathrm{GF}(q^m)\).

Examples

Find all primitive elements for the degree \(4\) extension of \(\mathrm{GF}(3)\).

In [1]: f = galois.conway_poly(3, 4); f
Out[1]: Poly(x^4 + 2x^3 + 2, GF(3))

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

Note from the list above that \(x + 2\) is a primitive element, but \(x + 1\) is not.

In [3]: galois.is_primitive_element("x + 2", f)
Out[3]: True

# x + 1 over GF(3) has integer equivalent of 4
In [4]: galois.is_primitive_element(4, f)
Out[4]: False

Last update: May 18, 2022