# Comparison of XL and Gröbner basis algorithms over Finite Fields

Abstract : This paper compares the XL algorithm with Gröbner basis algorithm. We explain the link between XL computation result and Gröbner basis with the well-known notion of $D$-Gröbner basis. Then we compare these algorithms in two cases: in the fields $\Fand$\Fwith $q\gg n$. For the field $\F we have proved that if XL needs to compute polynomial with degree$D$to terminate, the whole Gröbner basis is computated without exceeding the degree$D$. We have studied the XL algorithm and$F_5$algorithm on semi-regular sequences introduced in report~\cite{F5_complexite}. We show that the size of matrix constructed by XL is huge compared to the ones of$F_5$. So the complexity of XL is worth than$F_5$algorithm on these systems. For the field$\F we introduce an emulated algorithm using Gröbner basis computation to have a comparison between XL and Gröbner basis. We have proved that this algorithm will always reach a lower degree for intermediate polynomials than XL algorithm. A study on semi-regular sequences shows that $F_5$ always has a better behavior than XL algorithm especially when $m$ is near from $n$.
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https://hal.inria.fr/inria-00070747
Contributor : Rapport de Recherche Inria <>
Submitted on : Friday, May 19, 2006 - 9:29:59 PM
Last modification on : Thursday, March 21, 2019 - 12:59:08 PM
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• HAL Id : inria-00070747, version 1

### Citation

Jean-Charles Faugère, Gwénolé Ars. Comparison of XL and Gröbner basis algorithms over Finite Fields. [Research Report] RR-5251, INRIA. 2004, pp.26. ⟨inria-00070747⟩

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