Subquadratic-time factoring of polynomials over finite fields

Abstract

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n n over a finite field of constant cardinality in time O ( n 1.815 ) O(n^{1.815}) . Previous algorithms required time Θ ( n 2 + o ( 1 ) ) \Theta (n^{2+o(1)}) . The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n n over the finite field F q {\mathbb F}_q with q q elements, the algorithms use O ( n 1.815 log ⁡ q ) O(n^{1.815} \log q) arithmetic operations in F q {\mathbb F}_q . The new “baby step/giant step” techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.