Slimgb is a variation of Buchbergers’s algorithm for computing Gröbner bases in order to avoid intermediate coefficient swell. It is designed to keep coefficients small and polynomials short during the computation. This pays off in computation time as well as memory usage. One of the newly introduced concepts is a weighted length of a polynomial, being a combination of the number of terms, the ecart and the coefficient size and may depend on the ground field and the monomial ordering. Further key features of the algorithm are parallel reductions, exchanging members of the generating system for shorter intermediate results and an extended version of the product criterion. The algorithm is very flexible, the strategy is controlled by a single function which calculates the weighted length of a polynomial. All components of the algorithm depend on this function, hence one can easily customize the whole algorithm to fit the needs of specific problems by adjusting the weighted length. Its first implementation was made in Singular.

References in zbMATH (referenced in 15 articles , 1 standard article )

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  1. Faugère, Jean-Charles; Spaenlehauer, Pierre-Jean; Svartz, Jules: Sparse Gröbner bases: the unmixed case (2014)
  2. Bigatti, A.M.; Caboara, M.; Robbiano, L.: Computing inhomogeneous Gröbner bases (2011)
  3. Eder, Christian; Perry, John Edward: Signature-based algorithms to compute Gröbner bases (2011)
  4. Andres, Daniel; Brickenstein, Michael; Levandovskyy, Viktor; Martín-Morales, Jorge; Schönemann, Hans: Constructive $D$-module theory with Singular (2010)
  5. Brickenstein, Michael: Slimgb: Gröbner bases with slim polynomials (2010)
  6. Bulygin, Stanislav; Brickenstein, Michael: Obtaining and solving systems of equations in key variables only for the small variants of AES (2010)
  7. Eibach, Tobias; Völkel, Gunnar; Pilz, Enrico: Optimising Gröbner bases on Bivium (2010)
  8. Brickenstein, Michael; Dreyer, Alexander: Polybori: A framework for Gröbner-basis computations with Boolean polynomials (2009)
  9. Brickenstein, Michael; Dreyer, Alexander; Greuel, Gert-Martin; Wedler, Markus; Wienand, Oliver: New developments in the theory of Gröbner bases and applications to formal verification (2009)
  10. Suzuki, Akira: Computing Boolean Gröbner bases within linear algebra (2009)
  11. Suzuki, Akira: Computing Gröbner bases within linear algebra (2009)
  12. Albrecht, Martin: Algebraic attacks on the Courtois toy cipher (2008)
  13. Eibach, Tobias; Pilz, Enrico; Völkel, Gunnar: Attacking Bivium using SAT solvers (2008)
  14. King, Simon A.: Ideal Turaev-Viro invariants (2007)
  15. Levandovskyy, Viktor: Plural, a non-commutative extension of Singular: past, present and future. (2006)