# galois.primitive_poly¶

galois.primitive_poly(order, degree, method='min')

Returns a monic primitive polynomial $$f(x)$$ 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.

• method (str, optional) –

The search method for finding the primitive polynomial.

• "min" (default): Returns the lexicographically-minimal monic primitive polynomial.

• "max": Returns the lexicographically-maximal monic primitive polynomial.

• "random": Returns a randomly generated degree-$$m$$ monic primitive polynomial.

Returns

The degree-$$m$$ monic primitive polynomial 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

Notice galois.primitive_poly() returns the lexicographically-minimal primitive polynomial, whereas galois.conway_poly() returns the lexicographically-minimal primitive polynomial that is consistent with smaller Conway polynomials, which is not necessarily the same.

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

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

In [3]: galois.primitive_poly(7, 10)
Out[3]: Poly(x^10 + 5x^2 + x + 5, GF(7))

In [4]: galois.conway_poly(7, 10)
Out[4]: Poly(x^10 + x^6 + x^5 + 4x^4 + x^3 + 2x^2 + 3x + 3, GF(7))