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Programming and Computer Software

, Volume 45, Issue 8, pp 467–472 | Cite as

Minimal Basis of the Syzygy Module of Leading Terms

  • A. V. ShokurovEmail author
Article

Abstract

Systems of polynomial equations are one of the most universal mathematical objects. Almost all problems of cryptographic analysis can be reduced to solving systems of polynomial equations. The corresponding direction of research is called algebraic cryptanalysis. In terms of computational complexity, systems of polynomial equations cover the entire range of possible variants, from the algorithmic insolubility of Diophantine equations to well-known efficient methods for solving linear systems. Buchberger’s method [5] brings the system of algebraic equations to a system of a special type defined by the Gröbner original system of equations, which enables the elimination of dependent variables. The Gröbner basis is determined based on an admissible ordering on a set of terms. The set of admissible orderings on the set of terms is infinite and even continual. The most time-consuming step in finding the Gröbner basis by using Buchberger’s algorithm is to prove that all S-polynomials represent a system of generators of K[X]-module S-polynomials. Thus, a natural problem of finding this minimal system of generators arises. The existence of this system follows from Nakayama’s lemma. In this paper, we propose an algorithm for constructing this basis for any ordering.

Notes

REFERENCES

  1. 1.
    Gebauer, R. and Moller, H.M., On an installation of Buchberger’s algorithm, J. Symbolic Comput., 1987, no. 6, pp. 257–286.Google Scholar
  2. 2.
    Caboara, M., Kreuzer, M., and Robbiano, L., Efficiently computing minimal sets of critical pairs, J. Symbolic Comput., 2004, no. 38, pp. 1169–1190.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Lang, S., Algebra, Addison-Wesley, 1965.zbMATHGoogle Scholar
  4. 4.
    Agievich, S.V., Improved Buchberger algorithm, Proc. Inst. Math. Natl. Acad. Sci. Belarus, 2012, vol. 20, no. 1, pp. 3–13.zbMATHGoogle Scholar
  5. 5.
    Buchberger, B., Grobner bases: An algorithmic method in polynomial ideal, Multidimensional Systems Theory and Applications, 1985, pp. 184–232.zbMATHGoogle Scholar
  6. 6.
    Shokurov, A.V., On solving the systems of algebraic equations using Grobner bases, Tr. Inst. Sistemnogo Program. Ross. Akad. Nauk (Proc. Inst. Syst. Program. Russ. Acad. Sci.), 2013, vol. 25, pp. 195–206.Google Scholar
  7. 7.
    Varnovskiy, N.P., Zakharov, V.A., Kuzyurin, N.N., and Shokurov, A.V., The current state of art in program obfuscations: Definitions of obfuscation security, Program. Comput. Software, 2015, vol. 41, no. 6, pp. 361–372.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Varnovskiy, N.P., Martishin, S.A., Khrapchenko, M.V., and Shokurov, A.V., Secure cloud computing based on threshold homomorphic encryption, Program. Comput. Software, 2015, vol. 41, no. 4, pp. 215–218.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Varnovsky, N.P., Zakharov, V.A., and Shokurov, A.V., On the existence of provably secure cloud computing systems, Moscow Univ. Comput. Math. Cybernet., 2016, vol. 40, no. 2, pp. 83–88.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Varnovsky, N.P., Zakharov, V.A., and Shokurov, A.V., On the deductive security of queries to confidential databases in cloud computing systems, Moscow Univ. Comput. Math. Cybernet., 2017, vol. 41, no. 1, pp. 38–43.MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Ivannikov Institute for System Programming, Russian Academy of SciencesMoscowRussia

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