galois.primitive_element¶
- class 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 (galois.Poly) – The degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\) that defines the extension field \(\mathrm{GF}(p^m)\).
start (int, optional) – 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 (int, optional) – 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, optional) – 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
Examples
In [619]: GF = galois.GF(3) In [620]: f = galois.Poly([1,1,2], field=GF); f Out[620]: Poly(x^2 + x + 2, GF(3)) In [621]: galois.is_irreducible(f) Out[621]: True In [622]: galois.is_primitive(f) Out[622]: True In [623]: galois.primitive_element(f) Out[623]: Poly(x, GF(3))
In [624]: GF = galois.GF(3) In [625]: f = galois.Poly([1,0,1], field=GF); f Out[625]: Poly(x^2 + 1, GF(3)) In [626]: galois.is_irreducible(f) Out[626]: True In [627]: galois.is_primitive(f) Out[627]: False In [628]: galois.primitive_element(f) Out[628]: Poly(x + 1, GF(3))