## 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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