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Information, Security and Cryptology – ICISC 2009

Volume 5984 of the series Lecture Notes in Computer Science pp 87-100

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

  • Mohamed Saied Emam MohamedAffiliated withTU Darmstadt, FB Informatik
  • , Daniel CabarcasAffiliated withDepartment of Mathematical Sciences, University of Cincinnati, South China University of Technology
  • , Jintai DingAffiliated withDepartment of Mathematical Sciences, University of Cincinnati, South China University of Technology
  • , Johannes BuchmannAffiliated withTU Darmstadt, FB Informatik
  • , Stanislav BulyginAffiliated withCenter for Advanced Security Research Darmstadt (CASED)

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Abstract

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.

Keywords

Multivariate polynomial systems Gröbner basis XL algorithm Mutant MutantXL algorithm