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Fast Polynomial Factorization and Modular Composition

Kedlaya, Kiran S. and Umans, Christopher (2011) Fast Polynomial Factorization and Modular Composition. SIAM Journal on Computing, 40 (6). pp. 1767-1802. ISSN 0097-5397.

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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 [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187–224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179–1197]; 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 [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581–595], with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication [X. Huang and V. Y. Pan, J. Complexity, 14 (1998), pp. 257–299]. 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 near-optimally: 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 that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication.

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Additional Information:© 2011 Society for Industrial and Applied Mathematics. Received by the editors September 2, 2008; accepted for publication (in revised form) June 19, 2009; published electronically December 22, 2011. 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 [U08]. 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]. We thank Ariel Gabizon, Hendrik Hubrechts, Dieter Theunckens, and the anonymous referees for helpful comments, and Igor Sergeev for identifying an error in section 6. Finally, we thank Madhu Sudan for hosting a visit of the second author to MIT, which launched this collaboration.
Funding AgencyGrant Number
NSF DMS-0545904
Sloan Research FellowshipUNSPECIFIED
NSF CCF-0346991
Subject Keywords:modular composition; multivariate multipoint evaluation; multimodular reduction; polynomial factorization
Issue or Number:6
Classification Code:AMS Subject Headings: 11Y16, 13P05, 68W30, 68W40
Record Number:CaltechAUTHORS:20120125-151548395
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Official Citation:Fast Polynomial Factorization and Modular Composition Kiran S. Kedlaya and Christopher Umans, SIAM J. Comput. 40, 1767 (2011), DOI:10.1137/08073408X
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28972
Deposited By: Jason Perez
Deposited On:30 Jan 2012 16:01
Last Modified:03 Oct 2019 03:37

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