Factoring polynomials via relation-finding

In this paper we describe a new algorithm for fully factoring polynomials defined over the rationals or over number-fields. This algorithm uses as an essential subroutine, any fast relation finding algorithm for vectors of real numbers. Unlike previous algorithms which work on one factor at a time, the new algorithm finds all factors at once. Let P be a polynomial of degree n, height H(P) (=sum of the absolute values of P's coefficients), logarthmic height h(P)=log H(P). If we use the HJLS relation-finding algorithm of Hastad, Just, Lagarias and Schnorr, our algorithm has running time O(n5+Ch(P)) if fast multiplication is used, and O(n6+Ch(P)) if ordinary multiplication is used. This is an improvement by a factor of n over the algorithm of Schönhage, the previously best known.