Minimal Basis of the Syzygy Module of Leading Terms

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.

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Notes

  1. 1.

    In [4], at this point of proving the indecomposability of syzygies from Σ*, an incorrect assumption was made about the existence of a summand on the right-hand side of the representation of the syzygy whose highest term coincides with the highest term of the decomposable syzygy \({{S}_{{{{f}_{i}},{{f}_{j}}}}}\).

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.

  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.

  3. 3

    Lang, S., Algebra, Addison-Wesley, 1965.

    Google Scholar 

  4. 4

    Agievich, S.V., Improved Buchberger algorithm, Proc. Inst. Math. Natl. Acad. Sci. Belarus, 2012, vol. 20, no. 1, pp. 3–13.

    MATH  Google Scholar 

  5. 5

    Buchberger, B., Grobner bases: An algorithmic method in polynomial ideal, Multidimensional Systems Theory and Applications, 1985, pp. 184–232.

    Google 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.

  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.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google Scholar 

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Correspondence to A. V. Shokurov.

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Translated by Yu. Kornienko

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Shokurov, A.V. Minimal Basis of the Syzygy Module of Leading Terms. Program Comput Soft 45, 467–472 (2019). https://doi.org/10.1134/S036176881908005X

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