galois.primitive_polys¶

galois.primitive_polys(order, degree)¶

Returns all monic primitive polynomials \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\).

Parameters

order : int

The prime order \(q\) of the field \(\mathrm{GF}(q)\) that the polynomial is over.

degree : int

The degree \(m\) of the desired primitive polynomial.

Returns

All degree-\(m\) monic primitive polynomials over \(\mathrm{GF}(q)\).

Return type

galois.Poly

Notes

In addition to other applications, \(f(x)\) produces the field extension \(\mathrm{GF}(q^m)\) of \(\mathrm{GF}(q)\). Since \(f(x)\) is primitive, \(x\) is a primitive element \(\alpha\) of \(\mathrm{GF}(q^m)\) such that \(\mathrm{GF}(q^m) = \{0, 1, \alpha, \alpha^2, \dots, \alpha^{q^m-2}\}\).

Examples

All monic primitive polynomials over \(\mathrm{GF}(2)\) with degree \(5\).

In [1]: galois.primitive_polys(2, 5)
Out[1]: 
[Poly(x^5 + x^2 + 1, GF(2)),
 Poly(x^5 + x^3 + 1, GF(2)),
 Poly(x^5 + x^3 + x^2 + x + 1, GF(2)),
 Poly(x^5 + x^4 + x^2 + x + 1, GF(2)),
 Poly(x^5 + x^4 + x^3 + x + 1, GF(2)),
 Poly(x^5 + x^4 + x^3 + x^2 + 1, GF(2))]

All monic primitive polynomials over \(\mathrm{GF}(3^2)\) with degree \(2\).

In [2]: galois.primitive_polys(3**2, 2)
Out[2]: 
[Poly(x^2 + x + 3, GF(3^2)),
 Poly(x^2 + x + 7, GF(3^2)),
 Poly(x^2 + 2x + 3, GF(3^2)),
 Poly(x^2 + 2x + 7, GF(3^2)),
 Poly(x^2 + 3x + 6, GF(3^2)),
 Poly(x^2 + 3x + 7, GF(3^2)),
 Poly(x^2 + 4x + 5, GF(3^2)),
 Poly(x^2 + 4x + 6, GF(3^2)),
 Poly(x^2 + 5x + 3, GF(3^2)),
 Poly(x^2 + 5x + 5, GF(3^2)),
 Poly(x^2 + 6x + 6, GF(3^2)),
 Poly(x^2 + 6x + 7, GF(3^2)),
 Poly(x^2 + 7x + 3, GF(3^2)),
 Poly(x^2 + 7x + 5, GF(3^2)),
 Poly(x^2 + 8x + 5, GF(3^2)),
 Poly(x^2 + 8x + 6, GF(3^2))]