Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms
We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$...
Gespeichert in:
| Veröffentlicht in: | Groups, complexity, cryptology Jg. 14, Issue 1 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Episciences
01.01.2022
|
| Schlagworte: | |
| ISSN: | 1869-6104, 1869-6104 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | We study the problems of testing isomorphism of polynomials, algebras, and
multilinear forms. Our first main results are average-case algorithms for these
problems. For example, we develop an algorithm that takes two cubic forms $f,
g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are
isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has
direct practical implications, having been studied in multivariate cryptography
since the 1990s. Our second result concerns the complexity of testing
equivalence of alternating trilinear forms. This problem is of interest in both
mathematics and cryptography. We show that this problem is polynomial-time
equivalent to testing equivalence of symmetric trilinear forms, by showing that
they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore
is equivalent to testing isomorphism of cubic forms over most fields. |
|---|---|
| AbstractList | We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields. We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields. |
| Author | Grochow, Joshua A. Qiao, Youming Tang, Gang |
| Author_xml | – sequence: 1 givenname: Joshua A. surname: Grochow fullname: Grochow, Joshua A. – sequence: 2 givenname: Youming surname: Qiao fullname: Qiao, Youming – sequence: 3 givenname: Gang surname: Tang fullname: Tang, Gang |
| BookMark | eNp9kE9r3DAQxUXYQv4036AHf4DakWRJlnILS9MuBHppz2IsjxwttrVIamG_fe0khdBD5zKPYd6Px7smuyUuSMgnRhuhuNF3x9G5hlPOGyYa1hjRsgtyxbQytWJU7N7pS3Kb85Gu01ElKb0i9uE3JhixdpCxgmmMKZTnOVc-pqpgLmEZq5DjHNPpOeS5ir46xem8xDnAlD9vFuwTbGoZqvnXVMIUFoS0Eeb8kXzw6x_evu0b8vPxy4_9t_rp-9fD_uGpdq1qS71mg573puO846LFniF6CdxLaaiQA3hgBlVLte9AoHagDQXHBRrUoIb2hhxeuUOEoz2lMEM62wjBvhxiGi2kEtyEVnMGQjKnvKSi81K3ikqpBe0HI4ehW1n3ryyXYs4JvXWhQAlxKQnCZBm1L83brXm7NW-ZsMxuza9m8Y_5b5j_2v4ApFCLhA |
| CitedBy_id | crossref_primary_10_1007_s10623_024_01375_0 |
| ContentType | Journal Article |
| DBID | AAYXX CITATION DOA |
| DOI | 10.46298/jgcc.2022.14.1.9431 |
| DatabaseName | CrossRef DOAJ Directory of Open Access Journals |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1869-6104 |
| ExternalDocumentID | oai_doaj_org_article_821a451c6f5047f5836055840bd95dd7 10_46298_jgcc_2022_14_1_9431 |
| GroupedDBID | 0R~ 0~D 4.4 8FE 8FG AAFPC AAQCX AASOL AASQH AAWFC AAYXX ABAOT ABAQN ABIQR ABSOE ABUVI ABXMZ ACGFS ACZBO ADGQD ADGYE ADJVZ ADOZN AEJTT AEQDQ AEXIE AFBAA AFCXV AFQUK AIERV AJATJ ALMA_UNASSIGNED_HOLDINGS BAKPI BBCWN BCIFA BENPR BLHJL CFGNV CITATION GROUPED_DOAJ HZ~ IY9 J9A K6V O9- OK1 P2P P62 QD8 RDG SA. |
| ID | FETCH-LOGICAL-c363t-104ab2b97227243eb1eef5a2f559045dafa19e6308f7a4e8ca890ac24e9e8a6d3 |
| IEDL.DBID | DOA |
| ISICitedReferencesCount | 6 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000890601200001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 1869-6104 |
| IngestDate | Tue Oct 14 18:38:46 EDT 2025 Sat Nov 29 03:42:00 EST 2025 Tue Nov 18 22:12:27 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Language | English |
| License | https://arxiv.org/licenses/nonexclusive-distrib/1.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c363t-104ab2b97227243eb1eef5a2f559045dafa19e6308f7a4e8ca890ac24e9e8a6d3 |
| OpenAccessLink | https://doaj.org/article/821a451c6f5047f5836055840bd95dd7 |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_821a451c6f5047f5836055840bd95dd7 crossref_citationtrail_10_46298_jgcc_2022_14_1_9431 crossref_primary_10_46298_jgcc_2022_14_1_9431 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-01-01 |
| PublicationDateYYYYMMDD | 2022-01-01 |
| PublicationDate_xml | – month: 01 year: 2022 text: 2022-01-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationTitle | Groups, complexity, cryptology |
| PublicationYear | 2022 |
| Publisher | Episciences |
| Publisher_xml | – name: Episciences |
| SSID | ssj0000706500 |
| Score | 2.2551577 |
| Snippet | We study the problems of testing isomorphism of polynomials, algebras, and
multilinear forms. Our first main results are average-case algorithms for these... We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these... |
| SourceID | doaj crossref |
| SourceType | Open Website Enrichment Source Index Database |
| SubjectTerms | computer science - computational complexity computer science - data structures and algorithms |
| Title | Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms |
| URI | https://doaj.org/article/821a451c6f5047f5836055840bd95dd7 |
| Volume | 14, Issue 1 |
| WOSCitedRecordID | wos000890601200001&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: PRVAON databaseName: DOAJ Directory of Open Access Journals customDbUrl: eissn: 1869-6104 dateEnd: 20221231 omitProxy: false ssIdentifier: ssj0000706500 issn: 1869-6104 databaseCode: DOA dateStart: 20200101 isFulltext: true titleUrlDefault: https://www.doaj.org/ providerName: Directory of Open Access Journals |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV3LTtwwFLUQ6oIuKgqteLTIiy4JOH57SasiFhSxAImd5VeGqWYSNBkq8fe9NxlG0xWb7qLItqzjk_uIfY8J-ebAr2SdbFViIysZXagCSwxLnyDDtTqGOOjMXpubG_vw4G43rvrCM2GjPPAI3LnldZCqTrpRTJpGYdGBAq_JYnYq56GOnBm3kUwNNhh37xgba-Wk5s6e_54klCzkHIzDWX3mpKj_8UUbkv2Db7ncJR9WQSG9GCfzkWyVdo-8_7VWVO33ib8AzsG3XyVwOzTMJh1k9Y_znkLQSZcoldFO6LTv5h0AN-3ntGvoUzd7wapjYNgpdsE9YnxqMx3OEWKEGRY4wrz_RO4vf979uKpWlyNUSWixBPMpQ-TRGc4NlwJMbimNCryBFAHCtByaULuiBbONCbLYFKxjIXFZXLFBZ_GZbLddWw4IFSqZpHOMmQmZgo5GwBoZpVSqQ2HmkIhXmHxaKYfjBRYzDxnEAK5HcD2CC6mErz2Ce0iqda-nUTnjjfbfcQXWbVH3engBbPArNvi32HD0PwY5Jjs4tfFHyxeyvVw8l6_kXfqznPaLk4FofwHF3daS |
| linkProvider | Directory of Open Access Journals |
| 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=Average-case+algorithms+for+testing+isomorphism+of+polynomials%2C+algebras%2C+and+multilinear+forms&rft.jtitle=Groups%2C+complexity%2C+cryptology&rft.au=Joshua+A.+Grochow&rft.au=Youming+Qiao&rft.au=Gang+Tang&rft.date=2022-01-01&rft.pub=Episciences&rft.eissn=1869-6104&rft.volume=14%2C+Issue+1&rft_id=info:doi/10.46298%2Fjgcc.2022.14.1.9431&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_821a451c6f5047f5836055840bd95dd7 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1869-6104&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1869-6104&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1869-6104&client=summon |