A Caltech Library Service

Fast polynomial factorization and modular composition

Kedlaya, Kiran S. and Umans, Christopher (2008) Fast polynomial factorization and modular composition. In: Algebraic Methods in Computational Complexity: 08381 Abstracts Collection. Dagstuhl Publishing , Wadern, Germany, Art. No. 1777.

[img] PDF - Published Version
See Usage Policy.


Use this Persistent URL to link to this item:


We obtain randomized algorithms for factoring degree n univariate polynomials over F_q requiring O(^(n1.5+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 ≥ 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^((ω+1)/2)) field operations, where ω 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 α implies an algorithm for the other with exponent α + 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 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.

Item Type:Book Section
Related URLs:
URLURL TypeDescription
Additional Information:© 2008 Dagstuhl Publishing. The material in this paper appeared in conferences as [Uma08] and [KU08]. Supported by NSF DMS-0545904 (CAREER) and a Sloan Research Fellowship. Supported by NSF CCF-0346991 (CAREER), CCF-0830787, BSF 2004329, and a Sloan Research Fellowship. We thank Henry Cohn, Joachim von zur Gathen, David Harvey, Erich Kaltofen, and Eyal Rozenman for useful discussions, and Éric Schost for helpful comments on a draft of [Uma08]. We thank Swastik Kopparty and Madhu Sudan for some references mentioned in Section 5, and Ronald de Wolf and the FOCS 2008 referees for helpful comments on the conference paper [KU08]. Finally, we thank Madhu Sudan for hosting a visit of the second author to MIT, which launched this collaboration.
Funding AgencyGrant Number
Alfred P. Sloan FoundationUNSPECIFIED
Binational Science Foundation (USA-Israel)2004329
Subject Keywords:Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering
Record Number:CaltechAUTHORS:20191127-094213132
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100095
Deposited By: Tony Diaz
Deposited On:27 Nov 2019 18:16
Last Modified:27 Nov 2019 18:16

Repository Staff Only: item control page