Fast polynomial factorization and modular composition

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 & Shoup (1992) and Kaltofen & Shoup (1998)); for
$log q ge n$, 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 & 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 $alpha$ implies an
algorithm for the other with exponent $alpha + o(1)$, and vice
versa. We then give two new algorithms that solve the problem
optimally (up to lower order terms): an algebraic algorithm for
fields of characteristic at most $n^{o(1)}$, and a
nonalgebraic algorithm that works in arbitrary characteristic.
The latter algorithm works by lifting to characteristic 0,
applying a small number of rounds of {em 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.

Our algorithms use techniques which are commonly employed in
practice, so they may be competitive for real problem sizes. This
contrasts with all previous subquadratic algorithsm for these
problems, which rely on fast matrix multiplication.

This is joint work with Kiran Kedlaya.