{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article2237","name":"Fast polynomial factorization and modular composition","abstract":"We obtain randomized algorithms for factoring degree $n$\r\nunivariate polynomials over $F_q$ requiring $O(n^{1.5 +\r\no(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations.\r\nWhen $log q < n$, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for\r\n$log q ge n$, it matches the asymptotic running time of the best\r\nknown algorithms.\r\n\r\nThe improvements come from new algorithms for modular composition\r\nof degree $n$ univariate polynomials, which is the asymptotic\r\nbottleneck in fast algorithms for factoring polynomials over\r\nfinite fields. The best previous algorithms for modular\r\ncomposition use $O(n^{(omega + 1)\/2})$ field operations, where\r\n$omega$ is the exponent of matrix multiplication (Brent & Kung\r\n(1978)), with a slight improvement in the exponent achieved by\r\nemploying fast rectangular matrix multiplication (Huang & Pan\r\n(1997)).\r\n\r\nWe show that modular composition and multipoint evaluation of\r\nmultivariate polynomials are essentially equivalent, in the sense\r\nthat an algorithm for one achieving exponent $alpha$ implies an\r\nalgorithm for the other with exponent $alpha + o(1)$, and vice\r\nversa. We then give two new algorithms that solve the problem\r\noptimally (up to lower order terms): an algebraic algorithm for\r\nfields of characteristic at most $n^{o(1)}$, and a\r\nnonalgebraic algorithm that works in arbitrary characteristic.\r\nThe latter algorithm works by lifting to characteristic 0,\r\napplying a small number of rounds of {em multimodular reduction},\r\nand finishing with a small number of multidimensional FFTs. The\r\nfinal evaluations are reconstructed using the Chinese Remainder\r\nTheorem. As a bonus, this algorithm produces a very efficient data\r\nstructure supporting polynomial evaluation queries, which is of\r\nindependent interest.\r\n\r\nOur algorithms use techniques which are commonly employed in\r\npractice, so they may be competitive for real problem sizes. This\r\ncontrasts with all previous subquadratic algorithsm for these\r\nproblems, which rely on fast matrix multiplication.\r\n\r\nThis is joint work with Kiran Kedlaya.","keywords":"Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering","author":[{"@type":"Person","name":"Kedlaya, Kiran","givenName":"Kiran","familyName":"Kedlaya"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"position":5,"pageStart":1,"pageEnd":33,"dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kedlaya, Kiran","givenName":"Kiran","familyName":"Kedlaya"},{"@type":"Person","name":"Umans, Christopher","givenName":"Christopher","familyName":"Umans"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08381.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume717","volumeNumber":8381,"name":"Dagstuhl Seminar Proceedings, Volume 8381","dateCreated":"2008-12-11","datePublished":"2008-12-11","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article2237","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume717"}}}