Sparse FGLM algorithms
Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms...
Saved in:
| Published in: | Journal of symbolic computation Vol. 80; no. 3; pp. 538 - 569 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Ltd
01.05.2017
Elsevier |
| Subjects: | |
| ISSN: | 0747-7171, 1095-855X |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering.
The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over 60000. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments.
First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is O(D(N1+nlog(D)2)), where N1 is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Gröbner basis of I via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp–Massey–Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided.
Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove that its construction is free. With the asymptotic analysis of such sparsity, we are able to show that for generic systems the complexity above becomes O(6/nπD2+n−1n). |
|---|---|
| AbstractList | Given a zero-dimensional ideal $I \subset \kx$ of degree $D$, the transformation of the ordering of its \grobner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical {\sf FGLM} algorithm. Combing all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over $40000$. Such an implementation outperforms the {\sf Magma} and {\sf Singular} ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is $O(D(N_1+n\log (D)))$, where $N_1$ is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX \grobner basis of $\sqrt{I}$ via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp--Massey--Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove its construction is free. With the asymptotic analysis of such sparsity, we are able to show for generic systems the complexity above becomes $O(\sqrt{6/n \pi} D^{2+\frac{n-1}{n}})$. Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over 60000. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is O(D(N1+nlog(D)2)), where N1 is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Gröbner basis of I via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp–Massey–Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove that its construction is free. With the asymptotic analysis of such sparsity, we are able to show that for generic systems the complexity above becomes O(6/nπD2+n−1n). |
| Author | Mou, Chenqi Faugère, Jean-Charles |
| Author_xml | – sequence: 1 givenname: Jean-Charles surname: Faugère fullname: Faugère, Jean-Charles email: Jean-Charles.Faugere@inria.fr organization: Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d'Informatique de Paris 6 (LIP6), Équipe PolSys, 4 place Jussieu, 75005 Paris, France – sequence: 2 givenname: Chenqi surname: Mou fullname: Mou, Chenqi email: Chenqi.Mou@buaa.edu.cn organization: Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d'Informatique de Paris 6 (LIP6), Équipe PolSys, 4 place Jussieu, 75005 Paris, France |
| BackLink | https://inria.hal.science/hal-00807540$$DView record in HAL |
| BookMark | eNp9kE1Lw0AQQBepYFu9Ct569ZA4k-xmEzyVYlsh4kEFb8t2srEb0qTshoL_3i3Vi4eeZhjmzcebsFHXd4axO4QYAbOHJm48xUlIY5AxJOKCjREKEeVCfI7YGCSXkUSJV2zifQMABU_FmN2-7bXzZrZclS8z3X71zg7bnb9ml7Vuvbn5jVP2sXx6X6yj8nX1vJiXEaWSDxEVVBiATG4yjqgJsqxKCi2kTNIUwzUmzYkQsdpkdU2S8kJCLiAjnqAhnk7Z_WnuVrdq7-xOu2_Va6vW81IdawA5SMHhgKFXnnrJ9d47Uyuygx5s3w1O21YhqKMK1aigQh1VKJAqqAgk_iP_Vp1jHk-MCe8frHHKkzUdmco6Q4OqenuG_gG6E3U_ |
| CitedBy_id | crossref_primary_10_1016_j_jco_2020_101502 crossref_primary_10_1016_j_jco_2020_101499 crossref_primary_10_1016_j_jsc_2016_12_006 crossref_primary_10_1016_j_jsc_2024_102367 crossref_primary_10_1145_3338637_3338641 crossref_primary_10_1016_j_jalgebra_2022_03_002 crossref_primary_10_1016_j_jsc_2022_08_014 crossref_primary_10_1016_j_jsc_2019_10_011 crossref_primary_10_1016_j_jsc_2021_11_001 crossref_primary_10_1016_j_jsc_2019_09_001 crossref_primary_10_1016_j_jsc_2021_07_002 crossref_primary_10_1109_TIT_2020_3027751 crossref_primary_10_1007_s10473_022_0617_7 crossref_primary_10_1016_j_jsc_2019_07_010 crossref_primary_10_1145_3614408_3614411 |
| Cites_doi | 10.1007/s002000050114 10.1016/S0022-4049(99)00005-5 10.1215/S0012-7094-87-05517-7 10.1109/18.476246 10.1006/jsco.1996.0145 10.1006/jsco.1993.1051 10.1016/S0747-7171(08)80086-7 10.1016/S0747-7171(88)80033-6 10.1016/0890-5401(90)90039-K 10.1016/j.tcs.2007.10.003 10.1007/s00200-006-0015-8 10.1007/s00145-013-9158-5 10.1016/j.jsc.2012.03.007 10.1109/TIT.1986.1057137 10.1007/s002000050084 10.1109/18.179340 10.1016/j.jsc.2014.09.025 10.1016/0196-6774(80)90013-9 10.1016/0024-3795(89)90032-3 |
| ContentType | Journal Article |
| Copyright | 2016 Elsevier Ltd Distributed under a Creative Commons Attribution 4.0 International License |
| Copyright_xml | – notice: 2016 Elsevier Ltd – notice: Distributed under a Creative Commons Attribution 4.0 International License |
| DBID | AAYXX CITATION 1XC VOOES |
| DOI | 10.1016/j.jsc.2016.07.025 |
| DatabaseName | CrossRef Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access) |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science |
| EISSN | 1095-855X |
| EndPage | 569 |
| ExternalDocumentID | oai:HAL:hal-00807540v1 10_1016_j_jsc_2016_07_025 S0747717116300700 |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 1B1 1RT 1~. 1~5 29L 4.4 457 4G. 5GY 5VS 6I. 6OB 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO AAYFN ABAOU ABBOA ABEFU ABFNM ABJNI ABMAC ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACNNM ACRLP ACZNC ADBBV ADEZE ADFGL ADMUD AEBSH AEKER AENEX AEXQZ AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AIALX AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR ASPBG AVWKF AXJTR AZFZN BKOJK BLXMC CAG COF CS3 DM4 DU5 EBS EFBJH EFLBG EJD EO8 EO9 EP2 EP3 FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA GBOLZ HVGLF HZ~ IHE IXB J1W KOM LG5 M25 M41 MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SSV SSW SSZ T5K TN5 UPT WUQ XPP YQT ZMT ZU3 ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO ADVLN AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD 1XC VOOES |
| ID | FETCH-LOGICAL-c374t-c9c9e0067b6411ac066d29a5772331201e38cc111db6ffc7c89708506c421ec43 |
| ISICitedReferencesCount | 32 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000390832000002&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0747-7171 |
| IngestDate | Tue Oct 14 20:43:36 EDT 2025 Sat Nov 29 07:14:27 EST 2025 Tue Nov 18 22:22:40 EST 2025 Fri Feb 23 02:31:31 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Keywords | Gröbner bases Sparse matrix Change of ordering Zero-dimensional ideal BMS algorithm Wiedemann algorithm Zero-dimensional ideals Grobner bases |
| Language | English |
| License | Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c374t-c9c9e0067b6411ac066d29a5772331201e38cc111db6ffc7c89708506c421ec43 |
| OpenAccessLink | https://inria.hal.science/hal-00807540 |
| PageCount | 32 |
| ParticipantIDs | hal_primary_oai_HAL_hal_00807540v1 crossref_citationtrail_10_1016_j_jsc_2016_07_025 crossref_primary_10_1016_j_jsc_2016_07_025 elsevier_sciencedirect_doi_10_1016_j_jsc_2016_07_025 |
| PublicationCentury | 2000 |
| PublicationDate | May-June 2017 2017-05-00 2017-05 |
| PublicationDateYYYYMMDD | 2017-05-01 |
| PublicationDate_xml | – month: 05 year: 2017 text: May-June 2017 |
| PublicationDecade | 2010 |
| PublicationTitle | Journal of symbolic computation |
| PublicationYear | 2017 |
| Publisher | Elsevier Ltd Elsevier |
| Publisher_xml | – name: Elsevier Ltd – name: Elsevier |
| References | Feng, Rao (br0240) 1993; 39 Faugère, Gaudry, Huot, Renault (br0180) 2013; 27 Wang (br0430) 2001 Becker, Mora, Marinari, Traverso (br0050) 1994 Cox, Little, O'Shea (br0130) 1998 Von zur Gathen, Gerhard (br0420) 2003 Jonckheere, Ma (br0270) 1989; 125 Lazard (br0300) 1983 Buchmann, Pyshkin, Weinmann (br0110) 2006 Faugère, Safey El Din, Spaenlehauer (br0230) 2012 Basiri, Faugère (br0030) 2003 Faugère, Safey El Din, Spaenlehauer (br0220) 2010 Miller, Sturmfels (br0330) 2005; vol. 227 Berthomieu, Boyer, Faugère (br0070) 2015 Saints, Heegard (br0390) 2002; 41 Pascal, Schost (br0370) 2006 Bayer, Stillman (br0040) 1987; 55 Kaltofen, Pan (br0280) 1991 Buchberger (br0100) 1985 Galligo (br0250) 1974 Bardet (br0010) 2004 Lasserre, Laurent, Mourrain, Rostalski, Trébuchet (br0290) 2013; 51 Pardue (br0360) 1994 Wiedemann (br0440) 1986; 32 Becker, Weispfenning, Kredel (br0060) 1993 Bras-Amorós, O'Sullivan (br0080) 2006; 17 Lazard (br0310) 1992; 13 Faugère (br0170) 2002 Sakata (br0410) 1990; 84 Rouillier (br0380) 1999; 9 Faugère, Mou (br0210) 2011 Mou (br0350) 2012; 32 Morgan (br0340) 1987 Faugère, Gaudry, Huot, Renault (br0190) 2014 Bardet, Faugère, Salvy (br0020) 2015; 70 Høholdt, van Lint, Pellikaan (br0260) 1998 Loustaunau, York (br0320) 1997; 8 Brent, Gustavson, Yun (br0090) 1980; 1 Collart, Kalkbrener, Mall (br0120) 1997; 24 Sakata (br0400) 1988; 5 Eisenbud (br0150) 1995; vol. 150 Faugère (br0160) 1999; 139 Dahan, Jin, Moreno Maza, Schost (br0140) 2008; 392 Faugère, Gianni, Lazard, Mora (br0200) 1993; 16 Jonckheere (10.1016/j.jsc.2016.07.025_br0270) 1989; 125 Pardue (10.1016/j.jsc.2016.07.025_br0360) 1994 Lasserre (10.1016/j.jsc.2016.07.025_br0290) 2013; 51 Becker (10.1016/j.jsc.2016.07.025_br0050) 1994 Bras-Amorós (10.1016/j.jsc.2016.07.025_br0080) 2006; 17 Buchberger (10.1016/j.jsc.2016.07.025_br0100) 1985 Eisenbud (10.1016/j.jsc.2016.07.025_br0150) 1995; vol. 150 Sakata (10.1016/j.jsc.2016.07.025_br0410) 1990; 84 Kaltofen (10.1016/j.jsc.2016.07.025_br0280) 1991 Wiedemann (10.1016/j.jsc.2016.07.025_br0440) 1986; 32 Bardet (10.1016/j.jsc.2016.07.025_br0020) 2015; 70 Høholdt (10.1016/j.jsc.2016.07.025_br0260) 1998 Wang (10.1016/j.jsc.2016.07.025_br0430) 2001 Faugère (10.1016/j.jsc.2016.07.025_br0190) 2014 Pascal (10.1016/j.jsc.2016.07.025_br0370) 2006 Faugère (10.1016/j.jsc.2016.07.025_br0160) 1999; 139 Faugère (10.1016/j.jsc.2016.07.025_br0220) 2010 Faugère (10.1016/j.jsc.2016.07.025_br0230) 2012 Miller (10.1016/j.jsc.2016.07.025_br0330) 2005; vol. 227 Berthomieu (10.1016/j.jsc.2016.07.025_br0070) 2015 Lazard (10.1016/j.jsc.2016.07.025_br0300) 1983 Rouillier (10.1016/j.jsc.2016.07.025_br0380) 1999; 9 Buchmann (10.1016/j.jsc.2016.07.025_br0110) 2006 Bardet (10.1016/j.jsc.2016.07.025_br0010) 2004 Sakata (10.1016/j.jsc.2016.07.025_br0400) 1988; 5 Lazard (10.1016/j.jsc.2016.07.025_br0310) 1992; 13 Von zur Gathen (10.1016/j.jsc.2016.07.025_br0420) 2003 Galligo (10.1016/j.jsc.2016.07.025_br0250) 1974 Collart (10.1016/j.jsc.2016.07.025_br0120) 1997; 24 Faugère (10.1016/j.jsc.2016.07.025_br0180) 2013; 27 Brent (10.1016/j.jsc.2016.07.025_br0090) 1980; 1 Cox (10.1016/j.jsc.2016.07.025_br0130) 1998 Faugère (10.1016/j.jsc.2016.07.025_br0210) 2011 Morgan (10.1016/j.jsc.2016.07.025_br0340) 1987 Dahan (10.1016/j.jsc.2016.07.025_br0140) 2008; 392 Faugère (10.1016/j.jsc.2016.07.025_br0200) 1993; 16 Saints (10.1016/j.jsc.2016.07.025_br0390) 2002; 41 Bayer (10.1016/j.jsc.2016.07.025_br0040) 1987; 55 Mou (10.1016/j.jsc.2016.07.025_br0350) 2012; 32 Feng (10.1016/j.jsc.2016.07.025_br0240) 1993; 39 Faugère (10.1016/j.jsc.2016.07.025_br0170) 2002 Becker (10.1016/j.jsc.2016.07.025_br0060) 1993 Loustaunau (10.1016/j.jsc.2016.07.025_br0320) 1997; 8 Basiri (10.1016/j.jsc.2016.07.025_br0030) 2003 |
| References_xml | – volume: 9 start-page: 433 year: 1999 end-page: 461 ident: br0380 article-title: Solving zero-dimensional systems through the rational univariate representation publication-title: Appl. Algebra Eng. Commun. Comput. – year: 1994 ident: br0360 article-title: Nonstandard Borel-fixed ideals – start-page: 23 year: 2003 end-page: 29 ident: br0030 article-title: Changing the ordering of Gröbner bases with LLL: case of two variables publication-title: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation – volume: 39 start-page: 37 year: 1993 end-page: 45 ident: br0240 article-title: Decoding algebraic-geometric codes up to the designed minimum distance publication-title: IEEE Trans. Inf. Theory – year: 1987 ident: br0340 article-title: Solving Polynominal Systems Using Continuation for Engineering and Scientific Problems – volume: 13 start-page: 117 year: 1992 end-page: 131 ident: br0310 article-title: Solving zero-dimensional algebraic systems publication-title: J. Symb. Comput. – volume: 84 start-page: 207 year: 1990 end-page: 239 ident: br0410 article-title: Extension of the Berlekamp–Massey algorithm to publication-title: Inf. Comput. – volume: 139 start-page: 61 year: 1999 end-page: 88 ident: br0160 article-title: A new efficient algorithm for computing Gröbner bases ( publication-title: J. Pure Appl. Algebra – year: 1974 ident: br0250 article-title: A propos du théoreme de préparation de Weierstrass – volume: 17 start-page: 315 year: 2006 end-page: 335 ident: br0080 article-title: The correction capability of the Berlekamp–Massey–Sakata algorithm with majority voting publication-title: Appl. Algebra Eng. Commun. Comput. – start-page: 162 year: 2012 end-page: 169 ident: br0230 article-title: Critical points and Gröbner bases: the unmixed case publication-title: Proceedings of the 2012 International Symposium on Symbolic and Algebraic Computation – volume: 70 start-page: 49 year: 2015 end-page: 70 ident: br0020 article-title: On the complexity of the F5 Gröbner basis algorithm publication-title: J. Symb. Comput. – volume: 392 start-page: 37 year: 2008 end-page: 65 ident: br0140 article-title: Change of order for regular chains in positive dimension publication-title: Theor. Comput. Sci. – volume: 5 start-page: 321 year: 1988 end-page: 337 ident: br0400 article-title: Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array publication-title: J. Symb. Comput. – volume: 125 start-page: 65 year: 1989 end-page: 76 ident: br0270 article-title: A simple Hankel interpretation of the Berlekamp–Massey algorithm publication-title: Linear Algebra Appl. – start-page: 78 year: 2006 end-page: 88 ident: br0110 article-title: A zero-dimensional Gröbner basis for AES-128 publication-title: Fast Software Encryption – volume: 51 start-page: 63 year: 2013 end-page: 85 ident: br0290 article-title: Moment matrices, border bases and real radical computation publication-title: J. Symb. Comput. – year: 2001 ident: br0430 article-title: Elimination Methods – volume: 1 start-page: 259 year: 1980 end-page: 295 ident: br0090 article-title: Fast solution of Toeplitz systems of equations and computation of Padé approximants publication-title: J. Algorithms – year: 1998 ident: br0260 article-title: Algebraic Geometry Codes. Handbook of Coding Theory – volume: 41 start-page: 1733 year: 2002 end-page: 1751 ident: br0390 article-title: Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Gröbner bases publication-title: IEEE Trans. Inf. Theory – start-page: 184 year: 1985 end-page: 232 ident: br0100 article-title: Gröbner bases: an algorithmic method in polynomial ideal theory publication-title: Multidimensional Systems Theory – start-page: 61 year: 2015 end-page: 68 ident: br0070 article-title: Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences publication-title: Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation – start-page: 170 year: 2014 end-page: 177 ident: br0190 article-title: Sub-cubic change of ordering for Gröbner basis: a probabilistic approach publication-title: Proceedings of the 2014 International Symposium on Symbolic and Algebraic Computation – volume: 24 start-page: 465 year: 1997 end-page: 469 ident: br0120 article-title: Converting bases with the Gröbner walk publication-title: J. Symb. Comput. – start-page: 257 year: 2010 end-page: 264 ident: br0220 article-title: Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology publication-title: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation – volume: vol. 150 year: 1995 ident: br0150 article-title: Commutative Algebra: with a View Toward Algebraic Geometry publication-title: Graduate Texts in Mathematics – start-page: 180 year: 1991 end-page: 191 ident: br0280 article-title: Processor efficient parallel solution of linear systems over an abstract field publication-title: Proceedings of the Third Annual ACM Symposium on Parallel Algorithms and Architectures – year: 2003 ident: br0420 article-title: Modern Computer Algebra – volume: vol. 227 year: 2005 ident: br0330 article-title: Combinatorial Commutative Algebra publication-title: Graduate Texts in Mathematics – volume: 32 start-page: 2977 year: 2012 end-page: 2980 ident: br0350 article-title: Design of termination criterion of BMS algorithm for lexicographical ordering publication-title: J. Comput. Appl. – start-page: 129 year: 1994 end-page: 133 ident: br0050 article-title: The shape of the shape lemma publication-title: Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation – volume: 16 start-page: 329 year: 1993 end-page: 344 ident: br0200 article-title: Efficient computation of zero-dimensional Gröbner bases by change of ordering publication-title: J. Symb. Comput. – volume: 27 start-page: 595 year: 2013 end-page: 635 ident: br0180 article-title: Using symmetries in the index calculus for elliptic curves discrete logarithm publication-title: J. Cryptol. – volume: 55 start-page: 321 year: 1987 end-page: 328 ident: br0040 article-title: A theorem on refining division orders by the reverse lexicographic order publication-title: Duke Math. J. – year: 1993 ident: br0060 article-title: Gröbner Bases: a Computational Approach to Commutative Algebra publication-title: Graduate Texts in Mathematics – start-page: 75 year: 2002 end-page: 83 ident: br0170 article-title: A new efficient algorithm for computing Gröbner bases without reduction to zero ( publication-title: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation – volume: 8 start-page: 469 year: 1997 end-page: 483 ident: br0320 article-title: On the decoding of cyclic codes using Gröbner bases publication-title: Appl. Algebra Eng. Commun. Comput. – year: 2004 ident: br0010 article-title: Étude des systèmes algébriques surdéterminés. Applications aux codes correcteurs et à la cryptographie – year: 1998 ident: br0130 article-title: Using Algebraic Geometry – start-page: 146 year: 1983 end-page: 156 ident: br0300 article-title: Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations publication-title: Computer Algebra – start-page: 277 year: 2006 end-page: 284 ident: br0370 article-title: Change of order for bivariate triangular sets publication-title: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation – start-page: 115 year: 2011 end-page: 122 ident: br0210 article-title: Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices publication-title: Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation – volume: 32 start-page: 54 year: 1986 end-page: 62 ident: br0440 article-title: Solving sparse linear equations over finite fields publication-title: IEEE Trans. Inf. Theory – start-page: 184 year: 1985 ident: 10.1016/j.jsc.2016.07.025_br0100 article-title: Gröbner bases: an algorithmic method in polynomial ideal theory – volume: 9 start-page: 433 issue: 5 year: 1999 ident: 10.1016/j.jsc.2016.07.025_br0380 article-title: Solving zero-dimensional systems through the rational univariate representation publication-title: Appl. Algebra Eng. Commun. Comput. doi: 10.1007/s002000050114 – start-page: 61 year: 2015 ident: 10.1016/j.jsc.2016.07.025_br0070 article-title: Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences – volume: 139 start-page: 61 issue: 1–3 year: 1999 ident: 10.1016/j.jsc.2016.07.025_br0160 article-title: A new efficient algorithm for computing Gröbner bases (F4) publication-title: J. Pure Appl. Algebra doi: 10.1016/S0022-4049(99)00005-5 – year: 1998 ident: 10.1016/j.jsc.2016.07.025_br0260 – year: 1993 ident: 10.1016/j.jsc.2016.07.025_br0060 article-title: Gröbner Bases: a Computational Approach to Commutative Algebra – volume: 55 start-page: 321 issue: 2 year: 1987 ident: 10.1016/j.jsc.2016.07.025_br0040 article-title: A theorem on refining division orders by the reverse lexicographic order publication-title: Duke Math. J. doi: 10.1215/S0012-7094-87-05517-7 – volume: 41 start-page: 1733 issue: 6 year: 2002 ident: 10.1016/j.jsc.2016.07.025_br0390 article-title: Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Gröbner bases publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.476246 – year: 2004 ident: 10.1016/j.jsc.2016.07.025_br0010 – volume: 24 start-page: 465 issue: 3–4 year: 1997 ident: 10.1016/j.jsc.2016.07.025_br0120 article-title: Converting bases with the Gröbner walk publication-title: J. Symb. Comput. doi: 10.1006/jsco.1996.0145 – start-page: 180 year: 1991 ident: 10.1016/j.jsc.2016.07.025_br0280 article-title: Processor efficient parallel solution of linear systems over an abstract field – start-page: 75 year: 2002 ident: 10.1016/j.jsc.2016.07.025_br0170 article-title: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) – volume: 16 start-page: 329 issue: 4 year: 1993 ident: 10.1016/j.jsc.2016.07.025_br0200 article-title: Efficient computation of zero-dimensional Gröbner bases by change of ordering publication-title: J. Symb. Comput. doi: 10.1006/jsco.1993.1051 – year: 1998 ident: 10.1016/j.jsc.2016.07.025_br0130 – start-page: 170 year: 2014 ident: 10.1016/j.jsc.2016.07.025_br0190 article-title: Sub-cubic change of ordering for Gröbner basis: a probabilistic approach – start-page: 78 year: 2006 ident: 10.1016/j.jsc.2016.07.025_br0110 article-title: A zero-dimensional Gröbner basis for AES-128 – volume: 32 start-page: 2977 issue: 11 year: 2012 ident: 10.1016/j.jsc.2016.07.025_br0350 article-title: Design of termination criterion of BMS algorithm for lexicographical ordering publication-title: J. Comput. Appl. – volume: 13 start-page: 117 issue: 2 year: 1992 ident: 10.1016/j.jsc.2016.07.025_br0310 article-title: Solving zero-dimensional algebraic systems publication-title: J. Symb. Comput. doi: 10.1016/S0747-7171(08)80086-7 – start-page: 115 year: 2011 ident: 10.1016/j.jsc.2016.07.025_br0210 article-title: Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices – year: 1974 ident: 10.1016/j.jsc.2016.07.025_br0250 – year: 1987 ident: 10.1016/j.jsc.2016.07.025_br0340 – start-page: 257 year: 2010 ident: 10.1016/j.jsc.2016.07.025_br0220 article-title: Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology – volume: 5 start-page: 321 issue: 3 year: 1988 ident: 10.1016/j.jsc.2016.07.025_br0400 article-title: Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array publication-title: J. Symb. Comput. doi: 10.1016/S0747-7171(88)80033-6 – volume: 84 start-page: 207 issue: 2 year: 1990 ident: 10.1016/j.jsc.2016.07.025_br0410 article-title: Extension of the Berlekamp–Massey algorithm to N dimensions publication-title: Inf. Comput. doi: 10.1016/0890-5401(90)90039-K – year: 2001 ident: 10.1016/j.jsc.2016.07.025_br0430 – volume: 392 start-page: 37 year: 2008 ident: 10.1016/j.jsc.2016.07.025_br0140 article-title: Change of order for regular chains in positive dimension publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2007.10.003 – year: 1994 ident: 10.1016/j.jsc.2016.07.025_br0360 – volume: 17 start-page: 315 issue: 5 year: 2006 ident: 10.1016/j.jsc.2016.07.025_br0080 article-title: The correction capability of the Berlekamp–Massey–Sakata algorithm with majority voting publication-title: Appl. Algebra Eng. Commun. Comput. doi: 10.1007/s00200-006-0015-8 – volume: 27 start-page: 595 issue: 4 year: 2013 ident: 10.1016/j.jsc.2016.07.025_br0180 article-title: Using symmetries in the index calculus for elliptic curves discrete logarithm publication-title: J. Cryptol. doi: 10.1007/s00145-013-9158-5 – volume: vol. 150 year: 1995 ident: 10.1016/j.jsc.2016.07.025_br0150 article-title: Commutative Algebra: with a View Toward Algebraic Geometry – start-page: 23 year: 2003 ident: 10.1016/j.jsc.2016.07.025_br0030 article-title: Changing the ordering of Gröbner bases with LLL: case of two variables – volume: 51 start-page: 63 year: 2013 ident: 10.1016/j.jsc.2016.07.025_br0290 article-title: Moment matrices, border bases and real radical computation publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2012.03.007 – year: 2003 ident: 10.1016/j.jsc.2016.07.025_br0420 – volume: 32 start-page: 54 issue: 1 year: 1986 ident: 10.1016/j.jsc.2016.07.025_br0440 article-title: Solving sparse linear equations over finite fields publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1986.1057137 – volume: 8 start-page: 469 issue: 6 year: 1997 ident: 10.1016/j.jsc.2016.07.025_br0320 article-title: On the decoding of cyclic codes using Gröbner bases publication-title: Appl. Algebra Eng. Commun. Comput. doi: 10.1007/s002000050084 – start-page: 146 year: 1983 ident: 10.1016/j.jsc.2016.07.025_br0300 article-title: Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations – volume: 39 start-page: 37 issue: 1 year: 1993 ident: 10.1016/j.jsc.2016.07.025_br0240 article-title: Decoding algebraic-geometric codes up to the designed minimum distance publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.179340 – volume: 70 start-page: 49 year: 2015 ident: 10.1016/j.jsc.2016.07.025_br0020 article-title: On the complexity of the F5 Gröbner basis algorithm publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2014.09.025 – volume: 1 start-page: 259 issue: 3 year: 1980 ident: 10.1016/j.jsc.2016.07.025_br0090 article-title: Fast solution of Toeplitz systems of equations and computation of Padé approximants publication-title: J. Algorithms doi: 10.1016/0196-6774(80)90013-9 – volume: 125 start-page: 65 year: 1989 ident: 10.1016/j.jsc.2016.07.025_br0270 article-title: A simple Hankel interpretation of the Berlekamp–Massey algorithm publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(89)90032-3 – start-page: 129 year: 1994 ident: 10.1016/j.jsc.2016.07.025_br0050 article-title: The shape of the shape lemma – start-page: 277 year: 2006 ident: 10.1016/j.jsc.2016.07.025_br0370 article-title: Change of order for bivariate triangular sets – volume: vol. 227 year: 2005 ident: 10.1016/j.jsc.2016.07.025_br0330 article-title: Combinatorial Commutative Algebra – start-page: 162 year: 2012 ident: 10.1016/j.jsc.2016.07.025_br0230 article-title: Critical points and Gröbner bases: the unmixed case |
| SSID | ssj0009435 |
| Score | 2.38103 |
| Snippet | Given a zero-dimensional ideal I⊂K[x1,…,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial... Given a zero-dimensional ideal $I \subset \kx$ of degree $D$, the transformation of the ordering of its \grobner basis from DRL to LEX is a key step in... |
| SourceID | hal crossref elsevier |
| SourceType | Open Access Repository Enrichment Source Index Database Publisher |
| StartPage | 538 |
| SubjectTerms | BMS algorithm Change of ordering Computational Complexity Computer Science Data Structures and Algorithms Gröbner bases Sparse matrix Symbolic Computation Wiedemann algorithm Zero-dimensional ideal |
| Title | Sparse FGLM algorithms |
| URI | https://dx.doi.org/10.1016/j.jsc.2016.07.025 https://inria.hal.science/hal-00807540 |
| Volume | 80 |
| WOSCitedRecordID | wos000390832000002&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1095-855X dateEnd: 20180228 omitProxy: false ssIdentifier: ssj0009435 issn: 0747-7171 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1bT9swFLZG2cNeYGND46po2hPIUxzbtf1YITqGGJoEk_pmOY4LVCwtpEXw7zmOnZSLhrZJe4kqK3Gd80XH5-bvIPTZ5TnPQe1jJyXHzCrQg84oTAvS5Tkxiphh3WxCHB_LwUD9iO3dq7qdgChLeXurJv8VahgDsP3R2b-Au50UBuA3gA5XgB2ufwT8yQR8Vbfb_3r0fddcno3B-z-PlOTPjdDq7lfumYHr0vLZ47R838zO6jy6DHHqQ2dKHNPzLU7jWUjau_Lq4mEEAXaltl4vhLWaoy3zOqKqJrIWGHy9cJ8L2hHsMSw5HzxUn6ERU9R_PFC1xK2Uhy4sz7R0CBiMvowqTyJJAn1qOP_8hPz6xC_Dr4J4ajCRpgtoMRNcyQ5a7H3bHxzOCZZZaKbaLLvJYNe1fE_-6Hc2yMJ5E02vrYvTt2gpIpL0Apzv0CtXrqDlpuVGEjXwe7Qa0E08uskc3Q_oZ3__dO8Ax9YW2FLBptgqq5y3FPIuI8RYMPyKTBkOvg6lBJbqqLQW9iF_SnJohZVK1OSClmXEWUZXUaccl-4jSqjzNqJVjFPLCtc1mRC2KNTQmaygJl1DafOy2kbed99-5FI3BX4jDfLRXj46FRrks4Z22kcmgfTkpZtZI0EdrbZgjWmA-6XHPoG02-k9y_lB70j7Me_FCPAkbsj6v829gd7Mv_NN1Jlez9wWem1vphfV9Xb8cu4BNiBrDw |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Sparse+FGLM+algorithms&rft.jtitle=Journal+of+symbolic+computation&rft.au=Faug%C3%A8re%2C+Jean-Charles&rft.au=Mou%2C+Chenqi&rft.date=2017-05-01&rft.pub=Elsevier+Ltd&rft.issn=0747-7171&rft.eissn=1095-855X&rft.volume=80&rft.spage=538&rft.epage=569&rft_id=info:doi/10.1016%2Fj.jsc.2016.07.025&rft.externalDocID=S0747717116300700 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0747-7171&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0747-7171&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0747-7171&client=summon |