equaldegree

Equal-degree factorization, often abbreviated EDF, is a technique in computational algebra used to factor polynomials over finite fields when all irreducible factors have the same degree. It is not a stand-alone factoring method for arbitrary polynomials, but a specialized tool that serves as a subroutine within broader factorization algorithms, such as Cantor-Zassenhaus.

The central idea relies on the Frobenius automorphism in finite fields. If a polynomial f ∈ F_q[x] factorizes into irreducible factors all of degree d, then all roots of f lie in the extension field F_{q^d}. The Frobenius map α → α^q partitions these roots into cycles (orbits) of length d, with each orbit corresponding to one irreducible factor of f. EDF seeks to partition the roots into these orbits and hence to split f into its irreducible factors of degree d.

Conceptually, EDF constructs auxiliary polynomials or “test” functions that take constant values on roots within the same Frobenius orbit but take different values on roots from other orbits. By combining these tests with polynomial arithmetic modulo f and computing greatest common divisors, one obtains a nontrivial factor of f that corresponds to a union of orbits. Repeating the process on the factors yields the irreducible factors of f of degree d.

In practice, EDF is applied after a squarefree factorization step and is integrated into complete polynomial factorization algorithms over finite fields. Its efficiency depends on the field size, the common degree d, and the chosen testing strategies.