NDLI: Fast Modular Composition in any Characteristic

Content Provider	IEEE Xplore Digital Library
Author	Kedlaya, K.S. Umans, C.
Copyright Year	2008
Abstract	We give an algorithm for modular composition of degree n univariate polynomials over a finite field $F_{q}$ requiring n $^{1}$ $^{+}$ $^{o(1)}$ $log^{1}$ $^{+}$ $^{o(1)}$ q bit operations; this had earlier been achieved in characteristic $n^{o(1)}$ by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over $F_{q}$ requiring $(n^{1.5}$ $^{+}$ $^{o(1)}$ + n $^{1}$ $^{+}$ $^{o(1)}$ log q) $log^{1}$ $^{+}$ $^{o(1)}$ q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms.As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication.
Starting Page	146
Ending Page	155
File Size	333823
Page Count	10
File Format	PDF
ISBN	9780769534367
ISSN	02725428
DOI	10.1109/FOCS.2008.13
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-10-25
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Polynomials Galois fields Computer science Multidimensional systems Flexible printed circuits Mathematics Application software Testing Data structures Engineering profession multimodular reduction modular composition multipoint evaluation polynomial factorization
Content Type	Text
Resource Type	Article

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