MXL3: An Efficient Algorithm for Computing Gröbner Bases of Zero-Dimensional Ideals

  • Mohamed Saied Emam Mohamed
  • Daniel Cabarcas
  • Jintai Ding
  • Johannes Buchmann
  • Stanislav Bulygin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5984)


This paper introduces a new efficient algorithm, called MXL3, for computing Gröbner bases of zero-dimensional ideals. The MXL3 is based on XL algorithm, mutant strategy, and a new sufficient condition for a set of polynomials to be a Gröbner basis. We present experimental results comparing the behavior of MXL3 to F4 on HFE and random generated instances of the MQ problem. In both cases the first implementation of the MXL3 algorithm succeeds faster and uses less memory than Magma’s implementation of F4.


Multivariate polynomial systems Gröbner basis XL algorithm Mutant MutantXL algorithm 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mohamed Saied Emam Mohamed
    • 1
  • Daniel Cabarcas
    • 2
  • Jintai Ding
    • 2
  • Johannes Buchmann
    • 1
  • Stanislav Bulygin
    • 3
  1. 1.TU Darmstadt, FB InformatikDarmstadtGermany
  2. 2.Department of Mathematical SciencesUniversity of Cincinnati, South China University of Technology 
  3. 3.Center for Advanced Security Research Darmstadt (CASED) 

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