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On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra

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Abstract

Some techniques using linear algebra was introduced by Faugère in F4 to speed up the reduction process during Gröbner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Gröbner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function (given in F4). Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.

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Authors and Affiliations

  1. SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Yao Sun, Zhenyu Huang & Dongdai Lin

  2. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100090, China

    Dingkang Wang

Authors
  1. Yao Sun
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  2. Zhenyu Huang
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  3. Dongdai Lin
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  4. Dingkang Wang
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Corresponding author

Correspondence to Zhenyu Huang.

Additional information

This research is supported by the National Key Basic Research Program of China under Grant Nos. 2013CB834203 and 2011CB302400, the National Nature Science Foundation of China under Grant Nos. 11301523, 11371356, 61121062, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDA06010701, and IEE’s Research Project on Cryptography under Grant Nos. Y3Z0013102, Y3Z0018102, and Y4Z0061A02.

This paper was recommended for publication by Editor LI Ziming.

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Sun, Y., Huang, Z., Lin, D. et al. On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra. J Syst Sci Complex 29, 789–804 (2016). https://doi.org/10.1007/s11424-015-4085-1

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  • DOI: https://doi.org/10.1007/s11424-015-4085-1

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