galois.primitive_element¶

galois.primitive_element(irreducible_poly, start=None, stop=None, reverse=False)¶

Finds the smallest primitive element $g(x)$ of the Galois field $\mathrm{GF}(p^m)$ with degree-$m$ irreducible polynomial $f(x)$ over $\mathrm{GF}(p)$.

Parameters¶

irreducible_poly : Poly¶

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

start : Optional[int]¶

Starting value (inclusive, integer representation of the polynomial) in the search for a primitive element $g(x)$ of $\mathrm{GF}(p^m)$. The default is None which represents $p$, which corresponds to $g(x)=x$ over $\mathrm{GF}(p)$.

stop : Optional[int]¶

Stopping value (exclusive, integer representation of the polynomial) in the search for a primitive element $g(x)$ of $\mathrm{GF}(p^m)$. The default is None which represents $p^m$, which corresponds to $g(x)=x^m$ over $\mathrm{GF}(p)$.

reverse : bool¶

Search for a primitive element in reverse order, i.e. find the largest primitive element first. Default is False.

Returns¶

A primitive element of $\mathrm{GF}(p^m)$ with irreducible polynomial $f(x)$. The primitive element $g(x)$ is a polynomial over $\mathrm{GF}(p)$ with degree less than $m$.

Return type¶

Poly

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]: galois.primitive_element(f)
Out[5]: Poly(x, GF(3))

In [6]: GF = galois.GF(3)

In [7]: f = galois.Poly([1,0,1], field=GF); f
Out[7]: Poly(x^2 + 1, GF(3))

In [8]: galois.is_irreducible(f)
Out[8]: True

In [9]: galois.is_primitive(f)
Out[9]: False

In [10]: galois.primitive_element(f)
Out[10]: Poly(x + 1, GF(3))