Fast polynomial factorization and modular composition in small characteristic

Abstract

We obtain randomized algorithms for factoring degree n univariate polynomials over F_q that use O(n^(1.5 + o(1)) + n^(1 + o(1))log q) field operations, when the characteristic is at most n^(o(1)). When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q ≥ n, it matches the asymptotic running time of the best known algorithms. The improvements come from a new algorithm for modular composition of degree n univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use O(n^((ω+ 1)/2)) field operations, where ωis the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)). We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent α implies an algorithm for the other with exponent α + o(1), and vice versa. We then give a new algorithm that requires O(n^(1 + o(1))) field operations when the characteristic is at most n^(o(1)), which is optimal up to lower order terms. Our algorithms do not rely on fast matrix multiplication, in contrast to all previous subquadratic algorithms for these problems. The main operations are fast univariate polynomial arithmetic, multipoint evaluation, and interpolation, and consequently the algorithms could be feasible in practice.