Factoring Multivariate Polynomials over Finite Fields

We consider the deterministic complexity of the problem of polynomial factorization over finite fields given a finite field Fq and a polynomial h(x, y) ∈ Fq[x, y] compute the unique factorization of h(x, y) as a product of irreducible polynomials. This problem admits a randomized polynomial-time algorithm and no deterministic polynomial-time algorithm is known. In this chapter, we give a deterministic polynomial-time algorithm that partially factors the input polynomial h(x, y). The algorithm can be generalized to partially factor multivariate polynomials in an arbitary number of variables. We now describe precisely the output of our partial factoring algorithm. Associated with every Fq-irreducible factor f(x, y) of h(x, y) are two objects its total degree n and the smallest extension field Fqd of Fq over which f(x, y) splits into absolutely irreducible factors. Collecting all the Fq-irreducible factors of h(x, y) which have the same degree and the same splitting field, we get a unique factorization of h(x, y) into a product of “uniform polynomials” polynomials whose component Fq-irreducible factors all have the same degree and the same splitting field. It is this unique representation of h(x, y) as a product of uniform polynomials that is outputted by our algorithm.