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On functional decomposition of multivariate polynomials with differentiation and homogenization

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Abstract

This paper gives a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which were proposed by Ye, Dai, and Lam (1999) and were developed by Faugère, Perret (2006, 2008, 2009). The authors show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). The authors also propose a conjecture which asserts that the decomposition for a set of polynomials can be computed from that of its homogenization and show that the conjecture is valid with high probability for quartic polynomials. Finally, the authors prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space.

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References

  1. J. Patarin and L. Goubin, Asymmetric cryptography with S-boxes, Proceedings of ICICS’97, Lecture Notes in Computer Science, Springer, 1334, 1997.

  2. M. Dickerson, The functional decomposition of polynomials, Ph.D Thesis, TR 89-1023, Department of Compupter Science, Cornell University, Ithaca, NY, 1989.

    Google Scholar 

  3. J. Patarin and L. Goubin, Asymmetric cryptography with S-boxes (extended version), Available at http://citeseer.ist.psu.edu/patarin97asymmetric.html, 1997.

  4. J. Gutierre, R. Rubio, and J. von zur Gathen, Multivariate polynomial decomposition, Algebra in Engineering, Communication and Computing, 2003, 14(1): 11–31.

    MathSciNet  Google Scholar 

  5. J. von zur Gathen, Functional decomposition of polynomials: The tame case, Journal of Symbolic Computation, 1990, 9: 281–299.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. von zur Gathen, Functional decomposition of polynomials: The wild case, Journal of Symbolic Computation, 1990, 10: 437–452.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. W. Chionh, X. S. Gao, and L. Y. Shen, Inherently improper surface parametric supports, Computer Aided Geometric Design, 2006, 23: 629–639.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Li, L. Shen, and X. S. Gao, Proper reparametrization of rational ruled surface, Journal of Computer Science & Technology, 2008, 23(2): 290–297.

    Article  MathSciNet  Google Scholar 

  9. J. Li and X. S. Gao, The proper parametrization of a special class of rational parametric equations, Journal of Systems Science & Complexity, 2006, 19(3): 331–339.

    Article  MathSciNet  Google Scholar 

  10. L. Shen, E. Chionh, X. S. Gao, and J. Li, Proper reparametrization for inherently improper unirational vaeirties, Journal of Systems Science & Complexity, 2011, 24(2): 367–380.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. F. Ye, K. Y. Lam, and Z. D. Dai, Cryptanalysis of “2R” Schemes, Advances in Cryptology-CRYPTO 1999, Lecture Notes in Computer Science, Springer, 1999, 1666: 315–325.

  12. J. C. Faugère and L. Perret, Cryptanalysis of 2R − schemes, Advances in Cryptology-CRYPTO 2006, Lecture Notes in Computer Science, 2006, 4117: 357–372.

  13. J. C. Faugère and L. Perret, An efficient algorithm for decomposing multivariate ploynomials and its applications to cryptography, Special Issue of JSC, “Gröbner Bases techniques in Coding Theorey and Cryptography”, 2008.

  14. J. C. Faugère and L. Perret, High order derivatives and decomposition of multivariate polynomials, 2009, Proc. ISSAC, ACM Press, 2009.

  15. J. C. Faugère, J. von zur Gathen, and L. Perret, Decompostion of generic multivariate polynomials, ISSAC, 2010.

  16. D. Coppersmitha and S. Winograda, Matrix multiplication via arithmetic progressions, Journal of Symbolic Computation, 1990, 9(3): 251–280.

    Article  MathSciNet  Google Scholar 

  17. R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Company, 1983.

  18. D. Cox, J. Little, and D. O’shea, Ideals, Variaties, and Algorithms, 2nd ed., Springer-Verlag, 1996.

  19. T. Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, 1973.

  20. A. Schinzel, Polynomials wih Special Regard to Reducibility, Cambridge University Press, 2000.

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Authors and Affiliations

  1. Beijing Electronic Science and Technology Institute, Beijing, 100070, China

    Shangwei Zhao

  2. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ruyong Feng & Xiao-Shan Gao

Authors
  1. Shangwei Zhao
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  2. Ruyong Feng
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  3. Xiao-Shan Gao
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Additional information

This research is partially supported by a National Key Basic Research Project of China under Grant No. 2011CB302400 and by a Grant from NSFC with Nos 60821002 and 10901156.

This paper was recommended for publication by Editor Ziming LI.

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Zhao, S., Feng, R. & Gao, XS. On functional decomposition of multivariate polynomials with differentiation and homogenization. J Syst Sci Complex 25, 329–347 (2012). https://doi.org/10.1007/s11424-012-1144-8

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  • DOI: https://doi.org/10.1007/s11424-012-1144-8

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