research-article

Randomized Root Finding over Finite FFT-fields using Tangent Graeffe Transforms

Authors:

Joris van der Hoeven,

Grégoire LecerfAuthors Info & Claims

ISSAC '15: Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

Pages 197 - 204

https://doi.org/10.1145/2755996.2756647

Published: 24 June 2015 Publication History

Abstract

Consider a finite field F_q whose multiplicative group has smooth cardinality. We study the problem of computing all roots of a polynomial that splits over F_q, which was one of the bottlenecks for fast sparse interpolation in practice. We revisit and slightly improve existing algorithms and then present new randomized ones based on the Graeffe transform. We report on our implementation in the MATHEMAGIX computer algebra system, confirming that our ideas gain by a factor ten at least in practice, for sufficiently large inputs.

References

[1]

E. Bach. Comments on search procedures for primitive roots. Math. Comp., 66:1719--1727, 1997.

Digital Library

[2]

R. Berlekamp. Factoring polynomials over finite fields. Bell System Tech. J., 46:1853--1859, 1967.

[3]

R. Berlekamp. Factoring polynomials over large finite fields. Math. Comp., 24:713--735, 1970.

[4]

D. Bini and V. Y. Pan. Polynomial and matrix computations. Vol. 1. Fundamental algorithms. Progress in Theoretical Computer Science. Birkhuseär, 1994.

Digital Library

[5]

I. Bluestein. A linear filtering approach to the computation of discrete Fourier transform. IEEE Transactions on Audio and Electroacoustics, 18(4):451--455, 1970.

[6]

A. Bostan and É. Schost. Polynomial evaluation and interpolation on special sets of points. J. Complexity, 21(4):420--446, 2005.

Digital Library

[7]

G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Infor., 28(7):693--701, 1991.

Digital Library

[8]

G. Cantor and H. Zassenhaus. A new algorithm for factoring polynomials over finite fields. Math. Comp., 36(154):587--592, 1981.

[9]

PH. Flajolet and J.-M. Steyaert. A branching process arising in dynamic hashing, trie searching and polynomial factorization. In M. Nielsen and E. M. Schmidt, editors, Automata, Languages and Programming. Proceedings of the 9th ICALP Symposium, volume 140 of Lecture Notes in Comput. Sci., pages 239--251. Springer Berlin Heidelberg, 1982.

Digital Library

[10]

J. von zur Gathen. Factoring polynomials and primitive elements for special primes. Theoret. Comput. Sci., 52(1--2):77--89, 1987.

Digital Library

[11]

J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 2nd edition, 2003.

Digital Library

[12]

GCC, the GNU Compiler Collection. Software available at http://gcc.gnu.org, from 1987.

[13]

T. Granlund et al. GMP, the GNU multiple precision arithmetic library, from 1991. Software available at http://gmplib.org.

[14]

B. Grenet, J. van der Hoeven, and G. Lecerf. Deterministic root finding over finite fields using Graeffe transforms. http://hal.archives-ouvertes.fr/hal-01081743, 2015.

[15]

W. Hart, F. Johansson, and S. Pancratz. FLINT: Fast Library for Number Theory, 2014. Version 2.4.4, http://flintlib.org.

[16]

D. Harvey, J. van der Hoeven, and G. Lecerf. Even faster integer multiplication. http://arxiv.org/abs/1407.3360, 2014.

[17]

D. Harvey, J. van der Hoeven, and G. Lecerf. Faster polynomial multiplication over finite fields. http://arxiv.org/abs/1407.3361, 2014.

[18]

J. van der Hoeven et al. Mathemagix, from 2002. http://www.mathemagix.org.

[19]

J. van der Hoeven and G. Lecerf. Sparse polynomial interpolation in practice. ACM Commun. Comput. Algebra, 48(4), 2014. In section "ISSAC 2014 Software Presentations".

Digital Library

[20]

E. Kaltofen. Fifteen years after DSC and WLSS2, what parallel computations I do today. Invited lecture at PASCO 2010. In Proceedings of the 4th International Workshop on Parallel and Symbolic Computation, PASCO '10, pages 10--17. ACM Press, 2010.

Digital Library

[21]

E. Kaltofen and V. Shoup. Subquadratic-time factoring of polynomials over finite fields. Math. Comp., 67(223):1179--1197, 1998.

Digital Library

[22]

K. S. Kedlaya and C. Umans. Fast modular composition in any characteristic. In A. Z. Broder et al., editors, 49th Annual IEEE Symposium on Foundations of Computer Science 2008 (FOCS '08), pages 146--155. IEEE, 2008.

Digital Library

[23]

L. Kronecker. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. J. reine angew. Math., 92:1--122, 1882.

[24]

G. Malajovich and J. P. Zubelli. Tangent Graeffe iteration. Numer. Math., 89(4):749--782, 2001.

[25]

M. Mignotte and C. Schnorr. Calcul déterministe des racines d';un polynôme dans un corps fini. C. R. Acad. Sci. Paris Sér. I Math., 306(12):467--472, 1988.

[26]

T. Moenck. On the efficiency of algorithms for polynomial factoring. Math. Comp., 31:235--250, 1977.

[27]

G. L. Mullen and D. Panario. Handbook of Finite Fields. Discrete Mathematics and Its Applications. Chapman and Hall/CRC, 2013.

Digital Library

[28]

V. Pan. Solving a polynomial equation: Some history and recent progress. SIAM Rev., 39(2):187--220, 1997.

Digital Library

[29]

C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.

[30]

M. O. Rabin. Probabilistic algorithms in finite fields. SIAM J. Comput., 9(2):273--280, 1980.

Digital Library

[31]

L. Rónyai. Factoring polynomials modulo special primes. Combinatorica, 9(2):199--206, 1989.

[32]

V. Shoup. A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In S. M. Watt, editor, ISSAC '91: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, pages 14--21. ACM Press, 1991.

Digital Library

[33]

V. Shoup. Searching for primitive roots in finite fields. Math. Comp., 58: 369--380, 1992.

[34]

V. Shoup. NTL: A Library for doing Number Theory, 2014. Software, version 8.0.0. http://www.shoup.net/ntl.

Cited By

Demin Avan der Hoeven J(2025)Factoring sparse polynomials fastJournal of Complexity10.1016/j.jco.2025.101934(101934)Online publication date: Feb-2025
https://doi.org/10.1016/j.jco.2025.101934
van der Hoeven JLecerf G(2024)Fast interpolation of multivariate polynomials with sparse exponentsJournal of Complexity10.1016/j.jco.2024.101922(101922)Online publication date: Dec-2024
https://doi.org/10.1016/j.jco.2024.101922
van der Hoeven JLecerf G(2024)Sparse polynomial interpolation: faster strategies over finite fieldsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-024-00655-5Online publication date: 27-Apr-2024
https://doi.org/10.1007/s00200-024-00655-5
Show More Cited By

Index Terms

Randomized Root Finding over Finite FFT-fields using Tangent Graeffe Transforms
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations in finite fields
  2. Mathematical software

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cover image ACM Conferences

ISSAC '15: Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

June 2015

374 pages

ISBN:9781450334358

DOI:10.1145/2755996

Conference Chair:
Daniel Robertz
Plymouth University, United Kingdom
,
General Chair:
Steve Linton
University of St Andrews, United Kingdom
,
Program Chair:
Kazuhiro Yokoyama
Rikkyo University, Japan

Copyright © 2015 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 24 June 2015

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ISSAC'15: International Symposium on Symbolic and Algebraic Computation

July 6 - 9, 2015

Bath, United Kingdom

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ISSAC '15 Paper Acceptance Rate 43 of 71 submissions, 61%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Demin Avan der Hoeven J(2025)Factoring sparse polynomials fastJournal of Complexity10.1016/j.jco.2025.101934(101934)Online publication date: Feb-2025
https://doi.org/10.1016/j.jco.2025.101934
van der Hoeven JLecerf G(2024)Fast interpolation of multivariate polynomials with sparse exponentsJournal of Complexity10.1016/j.jco.2024.101922(101922)Online publication date: Dec-2024
https://doi.org/10.1016/j.jco.2024.101922
van der Hoeven JLecerf G(2024)Sparse polynomial interpolation: faster strategies over finite fieldsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-024-00655-5Online publication date: 27-Apr-2024
https://doi.org/10.1007/s00200-024-00655-5
van der Hoeven JMaza MZhi L(2022)On the Complexity of Symbolic ComputationProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3535493(3-12)Online publication date: 4-Jul-2022
https://dl.acm.org/doi/10.1145/3476446.3535493
van der Hoeven JMonagan M(2021)Computing one billion roots using the tangent Graeffe methodACM Communications in Computer Algebra10.1145/3457341.345734254:3(65-85)Online publication date: 15-Mar-2021
https://dl.acm.org/doi/10.1145/3457341.3457342
van der Hoeven JMonagan M(2020)Implementing the Tangent Graeffe Root Finding MethodMathematical Software – ICMS 202010.1007/978-3-030-52200-1_48(482-492)Online publication date: 13-Jul-2020
https://dl.acm.org/doi/10.1007/978-3-030-52200-1_48
Roche DKauers MOvchinnikov ASchost É(2018)What Can (and Can't) we Do with Sparse Polynomials?Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209027(25-30)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209027
Davenport JPetit CPring B(2017)A Generalised Successive Resultants AlgorithmArithmetic of Finite Fields10.1007/978-3-319-55227-9_9(105-124)Online publication date: 9-Mar-2017
https://doi.org/10.1007/978-3-319-55227-9_9
Grenet BHoeven JLecerf G(2016)Deterministic root finding over finite fields using Graeffe transformsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-015-0280-527:3(237-257)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/s00200-015-0280-5
Khochtali MRoche DTian XPernet C(2015)Parallel sparse interpolation using small primesProceedings of the 2015 International Workshop on Parallel Symbolic Computation10.1145/2790282.2790290(70-77)Online publication date: 10-Jul-2015
https://dl.acm.org/doi/10.1145/2790282.2790290

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