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An improvement over the GVW algorithm for inhomogeneous polynomial systems
Sun, Yao1; Huang, Zhenyu1; Wang, Dingkang2; Lin, Dongdai1
AbstractThe GVW algorithm provides a new framework for computing Grobner bases efficiently. If the input system is not homogeneous, some J-pairs with larger signatures but lower degrees may be rejected by GVW's criteria, and instead, GVW has to compute some J-pairs with smaller signatures but higher degrees. Consequently, degrees of polynomials appearing during the computations may unnecessarily grow up higher, and hence, the total computations become more expensive. This phenomenon happens more frequently when the coefficient field is a finite field and the field polynomials are involved in the computations. In this paper, a variant of the GVW algorithm, called M-GVW, is proposed. The concept of mutant pairs is introduced to overcome the inconveniences brought by inhomogeneous inputs. In aspects of implementations, to obtain efficient implementations of GVW/M-GVW over boolean polynomial rings, we take advantages of the famous library M4RI. We propose a new rotating swap method of adapting efficient routines in M4RI to deal with the one-direction reductions in GVW/M-GVW. Our implementations are tested with many examples from Boolean polynomial rings, and the timings show M-GVW usually performs much better than the original GVW algorithm if mutant pairs are found. (C) 2016 Elsevier Inc. All rights reserved.
KeywordGrobner basis The GVW algorithm Signature-based algorithm Linear algebra Boolean polynomial ring
Funding ProjectNational Key Basic Research Program of China[2013CB834203] ; National Natural Science Foundation of China[11301523] ; National Natural Science Foundation of China[61502485] ; Strategic Priority Research Program of the Chinese Academy of Sciences[XDA06010701] ; IEE's Research Project on Cryptography[Y4Z0061A02]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000381062900012
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Document Type期刊论文
Corresponding AuthorHuang, Zhenyu
Affiliation1.Chinese Acad Sci, Inst Informat Engn, SKLOIS, Beijing 100093, Peoples R China
2.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100190, Peoples R China
Recommended Citation
GB/T 7714
Sun, Yao,Huang, Zhenyu,Wang, Dingkang,et al. An improvement over the GVW algorithm for inhomogeneous polynomial systems[J]. FINITE FIELDS AND THEIR APPLICATIONS,2016,41:174-192.
APA Sun, Yao,Huang, Zhenyu,Wang, Dingkang,&Lin, Dongdai.(2016).An improvement over the GVW algorithm for inhomogeneous polynomial systems.FINITE FIELDS AND THEIR APPLICATIONS,41,174-192.
MLA Sun, Yao,et al."An improvement over the GVW algorithm for inhomogeneous polynomial systems".FINITE FIELDS AND THEIR APPLICATIONS 41(2016):174-192.
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