Fast polynomial factorization and modular composition

by Kiran Kedlaya and Chris Umans

Oded's comments

Just see the original abstract.

The original abstract

We obtain randomized algorithms for factoring degree n univariate polynomials over F_q requiring O(n^{1.5+o(1)}log^{1+o(1)}q+n^{1+o(1)}log^{2+o(1)}q) bit operations. When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen and Shoup (1992) and Kaltofen and Shoup (1998)); for log q > n, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms 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^{(\omega+1)/2}) field operations, where \omega is the exponent of matrix multiplication (Brent and Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang and 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 \alpha implies an algorithm for the other with exponent \alpha + o(1), and vice versa. We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most n^{o(1)}, and a nonalgebraic alorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese Remainder Theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. The new algorithms for modular composition lead to asymtotic improvements in algorithms for irreducibility testing, computing minimal polynomials, and other algebraic problems. Reducing the exponent of polynomial factorization from 1.5 to 1 is an outstanding open problem, and time permitting, I'll speculate on some possible routes to obtaining that result.

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