galois.primitive_polys¶

galois.primitive_polys(order: int, degree: int, reverse: bool = False) → Iterator[Poly]¶

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

Parameters

order: int¶

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

degree: int¶

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

reverse: bool = False¶

Indicates to return the primitive polynomials from lexicographically maximal to minimal. The default is False.

Returns

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

See also

Poly.is_primitive, irreducible_polys

Notes

If \(f(x)\) is a primitive polynomial over \(\mathrm{GF}(q)\) and \(a \in \mathrm{GF}(q) \backslash \{0\}\), then \(a \cdot f(x)\) is also primitive.

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

Return full list

All monic primitive polynomials over \(\mathrm{GF}(3)\) with degree \(4\). You may also use tuple() on the returned generator.

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

For loop

Loop over all the polynomials in reversed order, only finding them as needed. The search cost for the polynomials that would have been found after the break condition is never incurred.

In [2]: for poly in galois.primitive_polys(3, 4, reverse=True):
   ...:     if poly.coeffs[1] < 2:  # Early exit condition
   ...:         break
   ...:     print(poly)
   ...: 
x^4 + 2x^3 + 2x^2 + x + 2
x^4 + 2x^3 + x^2 + x + 2
x^4 + 2x^3 + 2

Manual iteration

Or, manually iterate over the generator.

In [3]: generator = galois.primitive_polys(3, 4, reverse=True); generator
Out[3]: <generator object primitive_polys at 0x7fb711af4ba0>

In [4]: next(generator)
Out[4]: Poly(x^4 + 2x^3 + 2x^2 + x + 2, GF(3))

In [5]: next(generator)
Out[5]: Poly(x^4 + 2x^3 + x^2 + x + 2, GF(3))

In [6]: next(generator)
Out[6]: Poly(x^4 + 2x^3 + 2, GF(3))