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- W2092435894 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 bestknown algorithms.The improvements come from new algorithms for modular compositionof degree $n$ univariate polynomials, which is the asymptoticbottleneck in fast algorithms for factoring polynomials overfinite fields. The best previous algorithms for modularcomposition 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 byemploying fast rectangular matrix multiplication (Huang & Pan(1997)).We show that modular composition and multipoint evaluation ofmultivariate polynomials are essentially equivalent, in the sensethat an algorithm for one achieving exponent $alpha$ implies analgorithm for the other with exponent $alpha + o(1)$, and viceversa. We then give two new algorithms that solve the problemoptimally (up to lower order terms): an algebraic algorithm forfields of characteristic at most $n^{o(1)}$, and anonalgebraic 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. Thefinal evaluations are reconstructed using the Chinese RemainderTheorem. As a bonus, this algorithm produces a very efficient datastructure supporting polynomial evaluation queries, which is ofindependent interest.Our algorithms use techniques which are commonly employed inpractice, so they may be competitive for real problem sizes. Thiscontrasts with all previous subquadratic algorithsm for theseproblems, which rely on fast matrix multiplication.This is joint work with Kiran Kedlaya." @default.
- W2092435894 created "2016-06-24" @default.
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- W2092435894 date "2008-01-01" @default.
- W2092435894 modified "2023-09-28" @default.
- W2092435894 title "Fast polynomial factorization and modular composition." @default.
- W2092435894 hasPublicationYear "2008" @default.
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