{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:26:37Z","timestamp":1787340397574,"version":"build-2736575974"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>We obtain randomized algorithms for factoring degree n univariate polynomials over $\\mathbb{F}_q$ requiring $O(n^{1.5 + o(1)}\\,{\\rm log}^{1+o(1)} q+ n^{1 + o(1)}\\,{\\rm log}^{2+o(1)} q)$ bit operations. When ${\\rm log}\\, q &lt; n$, this is asymptotically faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187\u2013224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179\u20131197]; for ${\\rm log}\\, q \\ge n$, 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 $O(n^{(\\omega + 1)\/2})$ field operations, where $\\omega$ is the exponent of matrix multiplication [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581\u2013595], 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\u2013299]. We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $\\alpha$ implies an algorithm for the other with exponent $\\alpha + o(1)$, 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.<\/jats:p>","DOI":"10.1137\/08073408x","type":"journal-article","created":{"date-parts":[[2011,12,22]],"date-time":"2011-12-22T10:05:42Z","timestamp":1324548342000},"page":"1767-1802","source":"Crossref","is-referenced-by-count":170,"title":["Fast Polynomial Factorization and Modular Composition"],"prefix":"10.1137","volume":"40","author":[{"given":"Kiran S.","family":"Kedlaya","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christopher","family":"Umans","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,12,22]]},"reference":[{"key":"R1","first-page":"7","volume":"5","author":"Belaga E. G.","year":"1961","journal-title":"Problemy Kibernet."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1970-0276200-X"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.1998.0216"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"D. J. Bernstein,\n                      Fast multiplication and its applications\n                      , in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, J. P. Buhler and P. Stevenhagen, eds., Cambridge University Press, Cambridge, UK, 2008, pp. 325\u2013384.","DOI":"10.1017\/9781139049801.011"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"A. Bostan, G. Lecerf, and E. Schost,\n                      Tellegen's principle into practice\n                      , in ISSAC '03: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, J. R. Sendra, ed., ACM, New York, 2003, pp. 37\u201344.","DOI":"10.1145\/860854.860870"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1145\/322092.322099"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"P. B\u00fcrgisser, M. Clausen, and M. A. Shokrollahi,\n                      Algebraic Complexity Theory\n                      , Grundlehren Math. Wiss. 315, Springer, Berlin, 1997.","DOI":"10.1007\/978-3-662-03338-8"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1981-0606517-5"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80013-2"},{"key":"R10","unstructured":"J.M. Couveignes and R. Lercier,\n                      Fast construction of irreducible polynomials over finite fields\n                      , http:\/\/arxiv.org\/abs\/0905.1642."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2007.02.047"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"V. Guruswami and A. Rudra,\n                      Explicit capacity-achieving list-decodable codes or decoding up to the singleton bound using Reed-Solomon codes\n                      , in STOC'06: Proceedings of the 38th Annual Symposium on Theory of Computing, J. M. Kleinberg, ed., ACM, New York, 2006, pp. 1\u201310.","DOI":"10.1145\/1132516.1132518"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1006\/jcom.1998.0476"},{"key":"R14","unstructured":"H. Hubrechts,\n                      \n                        Fast arithmetic in unramified\n                        p\n                        -adic fields\n                      \n                      , http:\/\/arxiv.org\/abs\/0906.5510."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-007-9000-2"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-004-0182-6"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"E. Kaltofen,\n                      Polynomial factorization: A success story\n                      , in ISSAC'03: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, J. R. Sendra, ed., ACM, New York, 2003, pp. 3\u20134.","DOI":"10.1145\/860854.860857"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"E. Kaltofen and V. Shoup,\n                      Fast polynomial factorization over high algebraic extensions of finite fields\n                      , in ISSAC'97: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 1997, pp. 184\u2013188.","DOI":"10.1145\/258726.258777"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-98-00944-2"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"K. S. Kedlaya and C. Umans,\n                      Fast modular composition in any characteristic\n                      , in FOCS'08: Proceedings of the 49th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2008, pp. 481\u2013490.","DOI":"10.1109\/FOCS.2008.13"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(95)80018-2"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"M. N\u00fcsken and M. Ziegler,\n                      Fast multipoint evaluation of bivariate polynomials\n                      , in Algorithms\u2014ESA, Lecture Notes in Comput. Sci. 3221, S. Albers and T. Radzik, eds., Springer, Berlin, 2004, pp. 544\u2013555.","DOI":"10.1007\/978-3-540-30140-0_49"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1070\/RM1966v021n01ABEH004147"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-008-9236-0"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"F. Parvaresh and A. 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Shoup,\n                      Efficient computation of minimal polynomials in algebraic extensions of finite fields\n                      , in ISSAC'99: Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 1999, pp. 53\u201358.","DOI":"10.1145\/309831.309859"},{"key":"R30","doi-asserted-by":"crossref","unstructured":"V. Shoup,\n                      A Computational Introduction to Number Theory and Algebra\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 2008.","DOI":"10.1017\/CBO9780511814549"},{"key":"R31","doi-asserted-by":"crossref","unstructured":"C. Umans,\n                      Fast polynomial factorization and modular composition in small characteristic\n                      , in STOC'08: Proceedings of the 40th Symposium on Theory of Computing, R. E. Ladner and C. 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