Flexible Partial Enlargement to Accelerate Gröbner Basis Computation over \(\mathbb{F}_2\)

  • Johannes Buchmann
  • Daniel Cabarcas
  • Jintai Ding
  • Mohamed Saied Emam Mohamed
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6055)


Recent developments in multivariate polynomial solving algorithms have made algebraic cryptanalysis a plausible threat to many cryptosystems. However, theoretical complexity estimates have shown this kind of attack unfeasible for most realistic applications. In this paper we present a strategy for computing Gröbner basis that challenges those complexity estimates. It uses a flexible partial enlargement technique together with reduced row echelon forms to generate lower degree elements–mutants. This new strategy surpasses old boundaries and obligates us to think of new paradigms for estimating complexity of Gröbner basis computation. The new proposed algorithm computed a Gröbner basis of a degree 2 random system with 32 variables and 32 equations using 30 GB which was never done before by any known Gröbner bases solver.


Algebraic cryptanalysis Gröbner basis Complexity HFE 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Johannes Buchmann
    • 1
  • Daniel Cabarcas
    • 2
  • Jintai Ding
    • 3
  • Mohamed Saied Emam Mohamed
    • 1
  1. 1.FB InformatikTU DarmstadtDarmstadtGermany
  2. 2.Department of Mathematical SciencesUniversity of Cincinnati 
  3. 3.Department of Mathematical SciencesUniversity of Cincinnati, South China University of Technology 

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