research-article

Supersparse black box rational function interpolation

Authors:

Erich L. Kaltofen,

Michael NehringAuthors Info & Claims

ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

Pages 177 - 186

https://doi.org/10.1145/1993886.1993916

Published: 08 June 2011 Publication History

Abstract

We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of non-zero terms and an upper bound on the degree. The result is found by interpolating the rational function modulo a small prime p, and then applying an effective version of Dirichlet's Theorem on primes in an arithmetic progression progressively lift the result to larger primes. Eventually we reach a prime number that is larger than the inputted degree bound and we can recover the original function exactly. In a variant, the initial prime p is large, but the exponents of the terms are known modulo larger and larger factors of p-1.

The algorithm, as presented, is conjectured to be polylogarithmic in the degree, but exponential in the number of terms. Therefore, it is very effective for rational functions with a small number of non-zero terms, such as the ratio of binomials, but it quickly becomes ineffective for a high number of terms.

The algorithm is oblivious to whether the numerator and denominator have a common factor. The algorithm will recover the sparse form of the rational function, rather than the reduced form, which could be dense. We have experimentally tested the algorithm in the case of under 10 terms in numerator and denominator combined and observed its conjectured high efficiency.

References

[1]

Candes, E., and Tao, T. Decoding by linear programming. IEEE Trans. Inf. Theory shape it-51, 12 (2005), 4203--4215.

Digital Library

[2]

Coleman, T., and Pothen, A. The null space problem I. complexity. SIAM. J. on Algebraic and Discrete Methods 7 (1986), 527--537.

Digital Library

[3]

Cucker, F., Koiran, P., and Smale, S. A polynomial time algorithm for diophantine equations in one variable. J. Symbolic Comput. 27, 1 (1999), 21--29.

Digital Library

[4]

Filaseta, M., Granville, A., and Schinzel, A. Irreducibility and greatest common divisor algorithms for sparse polynomials, 2007. Manuscript submitted.

[5]

Garg, S., and Schost, Éric. Interpolation of polynomials given by straight-line programs. Theoretical Computer Science 410, 27--29 (2009), 2659 -- 2662.

Digital Library

[6]

Giesbrecht, M., Kaltofen, E., and Lee, W. Algorithms for computing sparsest shifts of polynomials in power, Chebychev, and Pochhammer bases. J. Symbolic Comput. 36, 3--4 (2003), 401--424.

Digital Library

[7]

Giesbrecht, M., and Roche, D. S. Interpolation of shifted-lacunary polynomials. Computing Research Repository abs/0810.5685 (2008). URL: http://arxiv.org/abs/0810.5685.

Digital Library

[8]

Giesbrecht, M., and Roche, D. S. On lacunary polynomial perfect powers. In ISSAC 2008 (New York, N. Y., 2008), D. Jeffrey, Ed., ACM Press, pp. 103--110.

Digital Library

[9]

Giesbrecht, M., and Roche, D. S. Detecting lacunary perfect powers and computing their roots. Computing Research Repository abs/0901.1848 (2009).

Digital Library

[10]

Giesbrecht, M., Roche, D. S., and Tilak, H. Computing sparse multiples of polynomials. In Proc. Internat. Symp. on Algorithms and Computation (ISAAC 2010) (2010), p. to appear.

[11]

Heath-Brown, D. R. Almost-primes in arithmetic progressions and short intervals. Math. Proc. Camb. Phil. Soc. 83 (1978), 357--375.

[12]

Heath-Brown, D. R. Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc 3 (1992), 265--338.

[13]

Kaltofen, E. Greatest common divisors of polynomials given by straight-line programs. J. ACM 35, 1 (1988), 231--264.

Digital Library

[14]

Kaltofen, E. Unpublished article fragment, 1988. URL http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_ratint.pdf.

[15]

Kaltofen, E. Fifteen years after DSC and WLSS2 What parallel computations I do today {Invited lecture at PASCO 2010}. In PASCO'10 Proc. 2010 Internat. Workshop on Parallel Symbolic Comput. (New York, N. Y., July 2010), M. Moreno Maza and J.-L. Roch, Eds., ACM, pp. 10--17.

Digital Library

[16]

Kaltofen, E., and Koiran, P. Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields. In ISSAC MMVI Proc. 2006 Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2006), J.-G. Dumas, Ed., ACM Press, pp. 162--168.

Digital Library

[17]

Kaltofen, E., and Lee, W. Early termination in sparse interpolation algorithms. J. Symbolic Comput. 36, 3--4 (2003), 365--400. Special issue Internat. Symp. Symbolic Algebraic Comput. (ISSAC 2002). Guest editors: M. Giusti & L. M. Pardo.

Digital Library

[18]

Kaltofen, E., and Trager, B. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comput. 9, 3 (1990), 301--320.

Digital Library

[19]

Kaltofen, E., and Villard, G. On the complexity of computing determinants. Computational Complexity 13, 3--4 (2004), 91--130.

Digital Library

[20]

Kaltofen, E., and Yang, Z. On exact and approximate interpolation of sparse rational functions. In ISSAC 2007 Proc. 2007 Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2007), C. W. Brown, Ed., ACM Press, pp. 203--210.

Digital Library

[21]

Kaltofen, E., Yang, Z., and Zhi, L. On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms. In SNC'07 Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput. (New York, N. Y., 2007), J. Verschelde and S. M. Watt, Eds., ACM Press, pp. 11--17.

Digital Library

[22]

Kipnis, A., and Shamir, A. Cryptanalysis of the HFE public key cryptosystem by relinearization. In Proc. CRYPTO '99 (1999), M. J. Wiener, Ed., vol. 1666 of Lecture Notes in Computer Science, Springer, pp. 19--30.

[23]

Lenstra, Jr., H. W. Finding small degree factors of lacunary polynomials. In Number Theory in Progress (1999), K. Gy\Hory, H. Iwaniec, and J. Urbanowicz, Eds., vol. 1 Diophantine Problems and Polynomials, Stefan Banach Internat. Center, Walter de Gruyter Berlin/New York, pp. 267--276.

[24]

Mikawa, H. On primes in arithmetic progressions. Tsukuba J. Mathematics 25, 1 (2001), 121--153.

[25]

Olesh, Z., and Storjohann, A. The vector rational function reconstruction problems. In Proc. Waterloo Workshop on Computer Algebra: devoted to the 60th birthday of Sergei Abramov (WWCA) (2007), pp. 137--149.

[26]

Plaisted, D. A. New NP-hard and NP-complete polynomial and integer divisibility problems. Theoretical Comput. Sci. 13 (1984), 125--138.

[27]

Pohlig, C. P., and Hellman, M. E. An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. Inf. Theory shape it-24 (1978), 106--110.

Cited By

Giorgi PGrenet BPerret du Cray ARoche D(2024)Fast interpolation and multiplication of unbalanced polynomialsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669717(437-446)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669717
Giorgi PGrenet BPerret du Cray ARoche DMaza MZhi L(2022)Sparse Polynomial Interpolation and Division in Soft-linear TimeProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3536173(459-468)Online publication date: 4-Jul-2022
https://dl.acm.org/doi/10.1145/3476446.3536173
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
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Index Terms

Supersparse black box rational function interpolation
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms

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cover image ACM Conferences

ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

June 2011

372 pages

ISBN:9781450306751

DOI:10.1145/1993886

General Chair:
Éric Schost
The University of Western Ontario, Canada
,
Program Chair:
Ioannis Z. Emiris
University of Athens, Greece

Copyright © 2011 ACM.

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Published: 08 June 2011

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ISSAC '11: International Symposium on Symbolic and Algebraic Computation

June 8 - 11, 2011

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Cited By

Giorgi PGrenet BPerret du Cray ARoche D(2024)Fast interpolation and multiplication of unbalanced polynomialsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669717(437-446)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669717
Giorgi PGrenet BPerret du Cray ARoche DMaza MZhi L(2022)Sparse Polynomial Interpolation and Division in Soft-linear TimeProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3536173(459-468)Online publication date: 4-Jul-2022
https://dl.acm.org/doi/10.1145/3476446.3536173
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
Kaltofen EYang ZNagasaka KWinkler FSzanto A(2014)Sparse multivariate function recovery with a high error rate in the evaluationsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608637(280-287)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608637
Chattopadhyay AGrenet BKoiran PPortier NStrozecki YMonagan MCooperman GGiesbrecht M(2013)Factoring bivariate lacunary polynomials without heightsProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465932(141-148)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1145/2465506.2465932
Kaltofen EYang ZMonagan MCooperman GGiesbrecht M(2013)Sparse multivariate function recovery from values with noise and outlier errorsProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465524(219-226)Online publication date: 26-Jun-2013
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Comer MKaltofen EPernet Cvan der Hoeven Jvan Hoeij M(2012)Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input valuesProceedings of the 37th International Symposium on Symbolic and Algebraic Computation10.1145/2442829.2442852(138-145)Online publication date: 22-Jul-2012
https://dl.acm.org/doi/10.1145/2442829.2442852

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