Revisit Sparse Polynomial Interpolation Based on Randomized Kronecker Substitution

  • Qiao-Long Huang
  • Xiao-Shan GaoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


In this paper, a new reduction based interpolation algorithm for general black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce the black-box multivariate polynomial interpolation problem to the black-box univariate polynomial interpolation problem over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities in most cases when combining with various univariate interpolation algorithms. A modified univariate Ben-Or and Tiwari algorithm over the finite field is proposed, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-Or and Tiwari algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields.


Randomized Kronecker substitution Sparse polynomial interpolation Black-box Finite field Monte Carlo algorithm 


  1. 1.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Ng, E.W. (ed.) Symbolic and Algebraic Computation. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979). Scholar
  2. 2.
    Ben-Or, M., Tiwari, P.: A deterministic algorithm for sparse multivariate polynomial interpolation. In: Proceedings STOC 1988, pp. 301–309. ACM Press (1988)Google Scholar
  3. 3.
    Kaltofen, E.L., Lakshman, Y.N.: Improved sparse multivariate polynomial interpolation algorithms. In: Proceedings ISSAC 1988, pp. 467–474. Springer-Verlag (1988)Google Scholar
  4. 4.
    Zippel, R.: Interpolating polynomials from their values. J. Symb. Comp. 9, 375–403 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lakshman, Y.N., Saunders, B.D.: Sparse polynomial interpolation in nonstandard bases. SIAM J. Comput. 24(2), 387–397 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Klivans, A.R., Spielman, D.: Randomness efficient identity testing of multivariate polynomials. In: Proceedings STOC 2001, pp. 216–223. ACM Press (2001)Google Scholar
  7. 7.
    Grigoriev, D.Y., Karpinski, M., Singer, M.F.: Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM J. Comput. 19, 1059–1063 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Giesbrecht, M., Roche, D.S.: Diversification improves interpolation. In: Proceedings ISSAC 2011, pp. 123–130, ACM Press (2011)Google Scholar
  9. 9.
    Huang, M.D.A., Rao, A.J.: Interpolation of sparse multivariate polynomials over large finite fields with applications. J. Algorithms 33, 204–228 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Javadi, S.M.M., Monagan, M.: Parallel sparse polynomial interpolation over finite fields. In: Proceedings PASCO 2010, pp. 160–168. ACM Press (2010)Google Scholar
  11. 11.
    Kaltofen, E.L., Lee, W.S.: Early termination in sparse interpolation algorithms. J. Symbolic Comput. 36, 365–400 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hao, Z., Kaltofen, E.L., Zhi, L.: Numerical sparsity determination and early termination. In: Proceedings ISSAC 2017, pp. 247–254. ACM Press (2016)Google Scholar
  13. 13.
    Giesbrecht, M., Labahn, G., Lee, W.: Symbolic-numeric sparse interpolation of multivariate polynomials. In: Proceedings ISSAC 2006, pp. 116–123. ACM Press (2006)Google Scholar
  14. 14.
    Kaltofen, E.L., Lee, W.S., Yang, Z.: Fast estimates of hankel matrix condition numbers and numeric sparse interpolation. In: SNC 2011, pp. 130–136. ACM Press (2011)Google Scholar
  15. 15.
    Cuyt, A., Lee, W.S.: A new algorithm for sparse interpolation of multivariate polynomials. Theor. Comput. Sci. 409(2), 180–185 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Giesbrecht, M., Roche, D.S.: Interpolation of shifted-lacunary polynomials. Comput. Complex. 19(3), 333–354 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bläser, M., Jindal, G.: A new deterministic algorithm for sparse multivariate polynomial interpolation. In: Proceedings ISSAC 2014, pp. 51–58. ACM Press (2014)Google Scholar
  18. 18.
    Arnold, A., Giesbrecht, M., Roche, D.S.: Faster sparse multivariate polynomial interpolation of straight-line programs. J. Symbolic Comput. 75, 4–24 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Arnold, A., Giesbrecht, M., Roche, D.S.: Faster sparse interpolation of straight-line programs. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 61–74. Springer, Cham (2013). Scholar
  20. 20.
    Arnold, A., Roche, D.S.: Multivariate sparse interpolation using randomized Kronecker substitutions. In: ISSAC 2014, pp. 35–42. ACM Press (2014)Google Scholar
  21. 21.
    Garg, S., Schost, E.: Interpolation of polynomials given by straight-line programs. Theor. Comput. Sci. 410, 2659–2662 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Huang, Q.L., Gao, X.S.: Faster interpolation algorithms for sparse multivariate polynomials given by straight-Line programs. arXiv 1709.08979v4 (2017)Google Scholar
  23. 23.
    Huang, Q.L.: Sparse polynomial interpolation over fields with large or zero characteristic. In: Proceedings ISSAC 2019, ACM Press, to appear (2019)Google Scholar
  24. 24.
    Avendaño, M., Krick, T., Pacetti, A.: Newton-Hensel interpolation lifting. Found. Comput. Math. 6(1), 82–120 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Alon, N., Mansour, Y.: Epsilon-discrepancy sets and their application for interpolation of sparse polynomials. Inform. Process. Lett. 54(6), 337–342 (1995)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mansour, Y.: Randomized interpolation and approximation of sparse polynomials. SIAM J. Comput. 24(2), 357–368 (1995)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Arnold, A.: Sparse polynomial interpolation and testing. PhD Thesis, Waterloo Unversity (2016)Google Scholar
  28. 28.
    Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  29. 29.
    Gathen, J., von zur Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  30. 30.
    Pollard, J.M.: Monte Carlo Methods for Index Computation (mod \(p\)). Math. Comput. 32(143), 918–924 (1978)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.KLMM, UCAS, Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijingChina
  2. 2.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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