Factors the monic, square-free polynomial $f(x)$ of degree $rd$ into a product of $r$ irreducible factors with degree $d$.

Parameters:¶

degree: int¶

The degree $d$ of each irreducible factor of $f(x)$.

Returns:¶

The list of $r$ irreducible factors $\{g_1(x),\dots ,g_r(x)\}$ in lexicographical order.

Raises:¶

ValueError – If $f(x)$ is not monic, has degree 0, or is not square-free.

Notes

The Equal-Degree Factorization algorithm factors a square-free polynomial $f(x)$ with degree $rd$ into a product of $r$ irreducible polynomials each with degree $d$. This function implements the Cantor-Zassenhaus algorithm, which is probabilistic.

The Equal-Degree Factorization algorithm is often applied after the Distinct-Degree Factorization algorithm, see distinct_degree_factors(). A complete polynomial factorization is implemented in factors().

References

• Section 2.3 from https://people.csail.mit.edu/dmoshkov/courses/codes/poly-factorization.pdf

• Section 1 from https://www.csa.iisc.ac.in/~chandan/courses/CNT/notes/lec8.pdf

Examples

Factor a product of degree-1 irreducible polynomials over $\mathrm{GF}(2)$.

In [1]: a = galois.Poly([1, 0]); a, a.is_irreducible()
Out[1]: (Poly(x, GF(2)), True)

In [2]: b = galois.Poly([1, 1]); b, b.is_irreducible()
Out[2]: (Poly(x + 1, GF(2)), True)

In [3]: f = a * b; f
Out[3]: Poly(x^2 + x, GF(2))

In [4]: f.equal_degree_factors(1)
Out[4]: [Poly(x, GF(2)), Poly(x + 1, GF(2))]

Factor a product of degree-3 irreducible polynomials over $\mathrm{GF}(5)$.

In [5]: GF = galois.GF(5)

In [6]: a = galois.Poly([1, 0, 2, 1], field=GF); a, a.is_irreducible()
Out[6]: (Poly(x^3 + 2x + 1, GF(5)), True)

In [7]: b = galois.Poly([1, 4, 4, 4], field=GF); b, b.is_irreducible()
Out[7]: (Poly(x^3 + 4x^2 + 4x + 4, GF(5)), True)

In [8]: f = a * b; f
Out[8]: Poly(x^6 + 4x^5 + x^4 + 3x^3 + 2x^2 + 2x + 4, GF(5))

In [9]: f.equal_degree_factors(3)
Out[9]: [Poly(x^3 + 2x + 1, GF(5)), Poly(x^3 + 4x^2 + 4x + 4, GF(5))]