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Factoring high-degree polynomials by the black box Berlekamp algorithm

Authors:

Erich Kaltofen,

Austin LoboAuthors Info & Claims

ISSAC '94: Proceedings of the international symposium on Symbolic and algebraic computation

Pages 90 - 98

https://doi.org/10.1145/190347.190371

Published: 01 August 1994 Publication History

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References

[1]

Bach, E., Driscoll, J., and Shallit, J., "Factor refinement," J. Algorithms 15, pp. 199-222 (1993).

Digital Library

Google Scholar

[2]

Ben-Or, M., "Probabilistic algorithms in finite fields," Proc. 22nd IEEE Syrup. Foundations Comp. Sci., pp. 394-398 (1981).

Google Scholar

[3]

Berlekamp, E. R., "Factoring polynomials over finite fields," Bell Systems Tech. J. 46, pp. 1853-1859 (1967). Republished in revised form in: E. R. Berlekamp, Algebraic Coding Theory, Chapter 6, McGraw- Hill Publ., New York 1968.

Crossref

Google Scholar

[4]

Berlekamp, E. R., "Factoring polynomials over large finite fields," Math. Comp. 24, pp. 713-735 (1970).

Crossref

Google Scholar

[5]

Brent, R. P., Gustavson, F. G., and Yun, D. Y. Y., "Fast solution of Toeplitz systems of equations and computation of Pad~ approximants," J. Algorithms 1, pp. 259-295 (1980).

Crossref

Google Scholar

[6]

Butler, M. C. R., "On the reducibility of polynomials over a finite field," Quart. J. Math., Oxford Ser. (2) 5, pp. 102-107 (1954).

Crossref

Google Scholar

[7]

Camion, P., "Un algorithme de construction des idempotents primitifs d'ideaux d'algebres sur Fq," Ann. Discrete Math 12, pp. 55-63 (1982).

Google Scholar

[8]

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

Crossref

Google Scholar

[9]

Coppersmith, D., "Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm," Math. Comput. 62/205, pp. 333-350 (1994).

Digital Library

Google Scholar

[10]

Diaz, A., Hitz, M., Kaltofen, E., Lobo, A., and Valente, T., "Process scheduling in DSC and the large sparse linear systems challenge," in Proc. DISCO '93, Springer Lect. Notes Comput. Sci. 722, edited by A. Miola; pp. 66-80, 1993. Available from anonymous@fty, cs .rpi. edu in directory kaltofen.

Digital Library

Google Scholar

[11]

Dornstetter, J. L., "On the equivalence between Berlekamp's and Euclid's algorithms," IEEE Trans. Inf. Theory IT-33/3, pp. 428-431 (1987).

Digital Library

Google Scholar

[12]

Fleischmann, P., "Connections between the algorithms of Berlekamp and Niederreiter for factoring polynomials over {:q," Linear Algebra and Applications 192, pp. 01-108 (1993).

Crossref

Google Scholar

[13]

Gao, S. and von zur Gathen, J., "Berlekamp's and Niederreiter's polynomial factorization algorithm," in Finite Fields: Theory, Applications and Algorithms, Contemporary Mathematics, edited by L. Mullen and P. J.-S. Shiue; Amer. Math. Soc., Providence, Rhode Island, to appear, 1994.

Google Scholar

[14]

yon zur Gathen, J., "The polynomial factorization challenge," SIGSAM Bulletin 26/2, pp. 22-24 (1992).

Digital Library

Google Scholar

[15]

von zur Gathen, J. and Giesbrecht, M., "Constructing normal basis in finite fields," J. Symbolic Comput. 10/6, pp. 547-570 (1990).

Digital Library

Google Scholar

[16]

yon zur Gathen, J. and Shoup, V., "Computing Probenius maps and factoring polynomials," Comput. Complexity 2, pp. 187-224 (1992).

Crossref

Google Scholar

[17]

Kaltofen, E., "Sparse Hensel lifting," Proc. EUROCAL '85, Vol. 2, Springer Lec. Notes Comp. Sci. 204, pp. 4-17 (1985). Available from anonymouscftp. ~s .rpi. edu in directory kaltofen.

Digital Library

Google Scholar

[18]

Kaltofen, E., "Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems," in Proc. AAECC-IO, Springer Lect. Notes Comput. Sci. 673, edited by G. Cohen, T. Mora, and O. Moreno; pp. 195-212, 1993. Available from axtonymouscftp, cs .rpi. eduin directory kaltofen.

Digital Library

Google Scholar

[19]

Kaltofen, E., "Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems," Math. Comput., to appear (1995). Available from aaoaymou~cftp. ~s. ~pi. edu in directory kaltof en.

Digital Library

Google Scholar

[20]

Kaltofen, E., Musser, D. R., and Saunders, B. D., "A generalized class of polynomials that are hard to factor," SIAM J. Comp. 12/3, pp. 473-485 (1983).

Crossref

Google Scholar

[21]

Kaltofen, E. and Saunders, B. D., "On Wiedemann's method of solving sparse linear systems," in Proc. AAECC-9, Springer Lect. Notes Comput. Sci. 539; pp. 29-38, 1991. Available from anonymousOftp, cs. rpi. edu in directory kaltofen.

Digital Library

Google Scholar

[22]

Knuth, D. E., The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Ed. 2; Addison Wesley, Reading, MA, 1981.

Digital Library

Google Scholar

[23]

Massey, J. L., "Shift-register synthesis and BCH decoding," IEEE Trans. Inf. Theory IT-15, pp. 122-127 (1969).

Digital Library

Google Scholar

[24]

Niederreiter, H. and GSttfert, R., "Factorization of polynomials over finite fields and characteristic sequences," J. Symbolic Comput. 16/5, pp. 401-412 (1994).

Digital Library

Google Scholar

[25]

Patterson, D. A. and Hennessy, J. L., Computer Architecture A Quantitative Approach; Morgan Kaufmann Pub., San Matec, California, 1990.

Digital Library

Google Scholar

[26]

Rabin, M. O., "Probabilistic algorithms in finite fields," SIAM J. Comp. 9, pp. 273-280 (1980).

Crossref

Google Scholar

[27]

Schwarz, St., "On the reducibility of polynomials over a finite field," Quart. J. Math. Oxford Ser. (2) 7, pp. 110-124 (1956).

Crossref

Google Scholar

[28]

Shoup, V., "Factoring polynomials over finite fields" asymptotic complexity vs. reality," in Proc. Int. IMACS Syrup. on Symbolic Comput., New Trends and Developments; Lille, France, pp. 124-129, 1993.

Google Scholar

[29]

Wiedemann, D., "Solving sparse linear equations over finite fields," IEEE Trans. Inf. Theory IT-32, pp. 54- 62 (1986).

Digital Library

Google Scholar

Cited By

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Naudin PQuitté C(2019)Univariate polynomial factorization over finite fieldsTheoretical Computer Science10.1016/S0304-3975(97)80001-1191:1-2(1-36)Online publication date: 5-Jan-2019
https://dl.acm.org/doi/10.1016/S0304-3975%2897%2980001-1
von zur Gathen JGerhard J(2018)Polynomial factorization over F2Mathematics of Computation10.1090/S0025-5718-02-01421-771:240(1677-1698)Online publication date: 16-Dec-2018
https://dl.acm.org/doi/10.1090/S0025-5718-02-01421-7
Harasawa RSueyoshi YKudo A(2012)Improving the berlekamp algorithm for binomials x-aProceedings of the 4th international conference on Arithmetic of Finite Fields10.1007/978-3-642-31662-3_16(225-235)Online publication date: 16-Jul-2012
https://dl.acm.org/doi/10.1007/978-3-642-31662-3_16
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Index Terms

Factoring high-degree polynomials by the black box Berlekamp algorithm
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
      2. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices

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ISSAC '94: Proceedings of the international symposium on Symbolic and algebraic computation

August 1994

359 pages

ISBN:0897916387

DOI:10.1145/190347

Chairman:
Malcolm MacCallum
QMW, London, UK

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 ACM 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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Published: 01 August 1994

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ISSAC94: International Symposium on Symbolic and Algebraic Computation

July 20 - 22, 1994

Oxford, United Kingdom

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Naudin PQuitté C(2019)Univariate polynomial factorization over finite fieldsTheoretical Computer Science10.1016/S0304-3975(97)80001-1191:1-2(1-36)Online publication date: 5-Jan-2019
https://dl.acm.org/doi/10.1016/S0304-3975%2897%2980001-1
von zur Gathen JGerhard J(2018)Polynomial factorization over F2Mathematics of Computation10.1090/S0025-5718-02-01421-771:240(1677-1698)Online publication date: 16-Dec-2018
https://dl.acm.org/doi/10.1090/S0025-5718-02-01421-7
Harasawa RSueyoshi YKudo A(2012)Improving the berlekamp algorithm for binomials x-aProceedings of the 4th international conference on Arithmetic of Finite Fields10.1007/978-3-642-31662-3_16(225-235)Online publication date: 16-Jul-2012
https://dl.acm.org/doi/10.1007/978-3-642-31662-3_16
Hu WYang HHuang L(2012)Multidimensional Data Reduction Based on Compressed Sensing for Sensor NetworkFuture Wireless Networks and Information Systems10.1007/978-3-642-27323-0_90(721-728)Online publication date: 2012
https://doi.org/10.1007/978-3-642-27323-0_90
Boyer BDumas JGiorgi P(2010)Exact sparse matrix-vector multiplication on GPU's and multicore architecturesProceedings of the 4th International Workshop on Parallel and Symbolic Computation10.1145/1837210.1837224(80-88)Online publication date: 21-Jul-2010
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Tsuji K(2009)An improved EZ-GCD algorithm for multivariate polynomialsJournal of Symbolic Computation10.1016/j.jsc.2008.04.00644:1(99-110)Online publication date: 1-Jan-2009
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von zur Gathen JPanario D(2001)Factoring Polynomials Over Finite Fields: A SurveyJournal of Symbolic Computation10.1006/jsco.1999.100231:1-2(3-17)Online publication date: Jan-2001
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Díaz AKaltofen EWeispfenning VTrager B(1998)FOXBOXProceedings of the 1998 international symposium on Symbolic and algebraic computation10.1145/281508.281538(30-37)Online publication date: 1-Aug-1998
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Kaltofen EShoup V(1998)Subquadratic-time factoring of polynomials over finite fieldsMathematics of Computation10.1090/S0025-5718-98-00944-267:223(1179-1197)Online publication date: 1998
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Kaltofen EShoup VChar BWang P(1997)Fast polynomial factorization over high algebraic extensions of finite fieldsProceedings of the 1997 international symposium on Symbolic and algebraic computation10.1145/258726.258777(184-188)Online publication date: 1-Jul-1997
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Polynomials arising in factoring generalized Vandermonde determinants: an algorithm for computing their coefficients

Factoring non-monic polynomials represented by black boxes

Factoring bivariate lacunary polynomials without heights

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Polynomials arising in factoring generalized Vandermonde determinants: an algorithm for computing their coefficients

Factoring non-monic polynomials represented by black boxes

Factoring bivariate lacunary polynomials without heights

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