ABSTRACT
State of the art factoring in Q[x] is dominated in theory by a combinatorial reconstruction problem while, excluding some rare polynomials, performance tends to be dominated by Hensel lifting. We present an algorithm which gives a practical improvement (less Hensel lifting) for these more common polynomials. In addition, factoring has suffered from a 25 year complexity gap because the best implementations are much faster in practice than their complexity bounds. We illustrate that this complexity gap can be closed by providing an implementation which is comparable to the best current implementations and for which competitive complexity results can be proved.
- J. Abbott. Bounds on factors in Z{x}. arXiv:0904.3057, 2009.Google Scholar
- John Abbott, Victor Shoup, and Paul Zimmermann. Factorization in Z{x}: the searching phase. In Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, ISSAC '00, pages 1--7, New York, NY, USA, 2000. ACM. Google ScholarDigital Library
- Karim Belabas. A relative van Hoeij algorithm over number fields. Journal of Symbolic Computation, 37(5):641--668, 2004.Google ScholarCross Ref
- Karim Belabas, Mark van Hoeij, Jürgen Klüners, and Allan Steel. Factoring polynomials over global fields. Journal de Théorie des Nombres de Bordeaux, 21:15--39, 2009.Google ScholarCross Ref
- J. von zur Gathen and J. Gerhardt. Modern Computer Algebra, 2nd edition. Cambridge University Press, 2003. pages 235--242, 432--437. Google ScholarDigital Library
- W. Hart. FLINT. open-source C-library http://www.flintlib.org.Google Scholar
- W. Hart, M. v. Hoeij, and A. Novocin. Complexity analysis of factoring polynomials. http://andy.novocin.com/pro/complexity.pdf, 2010.Google Scholar
- Mark Van Hoeij. Factoring polynomials and the knapsack problem. Journal of Number Theory, 95:167--189, 2002.Google ScholarCross Ref
- E. Kaltofen. Factorization of polynomials. In Computing, Suppl. 4, pages 95--113. Springer-Verlag, 1982.Google Scholar
- E. Kaltofen. On the complexity of finding short vectors in integer lattices. In Proceedings of European Conference on Computer Algebra 1983 EUROCAL'83, volume 162 of Lecture Notes in Computer Science, pages 236--244. Springer-Verlag, 1983. Google ScholarDigital Library
- Erich Kaltofen, David R. Musser, and B. David Saunders. A generalized class of polynomials that are hard to factor. SIAM J. Comput., 12(3):473--483, 1983.Google ScholarCross Ref
- A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.Google ScholarCross Ref
- L. Lovász. An Algorithmic Theory of Numbers, Graphs and Convexity. Society for Industrial and Applied Mathematics (SIAM), 1986. (Conference Board of the Mathematical Sciences and National Science Foundarion) CBMS-NSF Regional Conference Series in Applied Mathematics.Google Scholar
- D. Micciancio. The shortest vector problem is NP-hard to approximate to within some constant. Society for Industrial and Applied Mathematics (SIAM) Journal on Computing, 30(6):2008--2035, 2001. Google ScholarDigital Library
- P. Q. Nguyen and D. Stehlé. Floating-point LLL revisited. In Proceedings of Eurocrypt 2005, volume 3494 of Lecture Notes in Computer Science, pages 215--233. Springer-Verlag, 2005. Google Scholar
- Phong Q. Nguyen and Damien Stehlé. LLL on the average. In Florian Hess, Sebastian Pauli, and Michael E. Pohst, editors, ANTS, volume 4076 of Lecture Notes in Computer Science, pages 238--256. Springer, 2006. Google ScholarDigital Library
- A. Novocin. Factoring Univariate Polynomials over the Rationals. PhD thesis, Florida State University, 2008. Google ScholarDigital Library
- C. P. Schnorr. A more efficient algorithm for lattice basis reduction. Journal of Algorithms, 9(1):47--62, 1988. Google ScholarDigital Library
- A. Schönhage. Factorization of univariate integer polynomials by Diophantine approximation and improved basis reduction algorithm. In Proceedings of the 1984 International Colloquium on Automata, Languages and Programming (ICALP 1984), volume 172 of Lecture Notes in Computer Science, pages 436--447. Springer-Verlag, 1984. Google ScholarDigital Library
- A. Storjohann. Faster Algorithms for Integer Lattice Basis Reduction. Technical Report TR249, Swiss Federal Institute of Technology Zürich, Department of Computer Science, 1996.Google Scholar
- Mark van Hoeij and Andrew Novocin. Gradual sub-lattice reduction and a new complexity for factoring polynomials. In LATIN, pages 539--553, 2010. Google ScholarDigital Library
- H. Zassenhaus. On Hensel Factorization I. In J. Number Theory, number 1, pages 291--311, 1969.Google Scholar
Index Terms
- Practical polynomial factoring in polynomial time
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