[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83582-en":3,"doc-seo-83582-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83582,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Extensions of Root-Based Attacks Against PLWE via Isomorphisms of Rings","The work studies how root-based attacks on PLWE can be extended. It constructs explicit isomorphisms from vulnerable instances to relate fully-split polynomial rings to polynomial rings where prior attacks apply, and proves formally that this does not introduce new vulnerabilities. Under such an isomorphism, sample distortion prevents the transformed samples from being distinguishable. The paper also shows that any isomorphism between fully-split polynomial rings must equal the constructed one, and generalizes beyond fully-split cases, revealing the same distortion for any field-extension degree.","arXiv :2607 .0 1340v2 [ cs .CR] 13 Jul 2026  \nExtensions of root-based attacks against PLWE via isomorphisms of rings  \nIván Blanco Chacón1 , Raúl Durán Díaz2 , and Rodrigo Martín Sánchez-Ledesma3 ,4  \n1 Departamento de Física y Matemáticas, Universidad de Alcalá, Spain  \n[ivan.blancoc@uah.es](ivan.blancoc@uah.es)  \n2 Departamento de Automática, Universidad de Alcalá, Spain  \n[raul.duran@uah.es](raul.duran@uah.es)  \n3 Departamento de Álgebra, Universidad Complutense de Madrid, Spain  \n[rodrma01@ucm.es](rodrma01@ucm.es)  \n4 Indra Sistemas de Comunicaciones Seguras, Spain  \nrmsanchezledesma@indra .es  \nAbstract. In the present work we address some key questions regarding the generalization of root-based attacks presented in [2] . In particular, we analyze potential root-based attacks extensions via the construction of explicit isomorphisms from vulnerable instances, and provide a formal proof that this approach will not yield any new vulnerabilities. To do so, we first construct an explicit isomorphism between fully-split polynomial rings and polynomial rings where previous attacks apply and show that the application of such an isomorphism will always distort the samples in a way that the resulting samples cannot be used to distinguish. Then, we prove that any isomorphism between fully-split polynomial rings must be of the form of the constructed isomorphism. Lastly, we study the generalization beyond fully split settings and show that the same distortion happens, regardless of the degree of the field extension.  \nKeywords: PLWE · Root-based attacks · Isomorphisms  \n1 Introduction  \nThe Ring Learning With Errors problem (RLWE) and the Polynomial Learning With Errors problem (PLWE) have become the most important paradigms in the quest of achieving postquantum security. Some of the strongest theoretical clues pointing in this direction are the worst case-average case reductions from an approximate version of the Shortest Vector Problem on ideal lattices to the RLWE problem, established in [7], and to the PLWE problem over power-of-two cyclotomic polynomials, established in [10] .  \nAn important debate in cryptography in general, and post-quantum cryptography in particular, is the one contrasting efficiency with security. Cryptographic schemes must obviously be secure, but they need also be practical and usable. This dichotomy is not a stranger to lattice-based cryptography, of which both PLWE and RLWE are representatives.  \nWithin lattices, this dilemma is represented by the definition of both unstructured and structured schemes. By structured, it is meant that additional algebraic structure is added to the scheme, whereas unstructured schemes normally rely on lighter mathematical constructions.  \nThe definition of these structured paradigms permits cryptographic schemes employing such structures to enjoy a number of privileges, most notably related to better performances and smaller cryptographic sizes. But the cost of doing so is the addition of the aforementioned heavy algebraic structure, which could potentially turn out to be the source of new attacks, and gives rise to some important questions: are there attacks that can effectively target this additional structure? Do they improve the best known attacks against LWE?  \nIt was precisely this line of thought that the works of [5,6] explored. In them, the authors introduce a general framework to exploit the algebraic structure of PLWE schemes in order to mount decisional attacks. In particular, evaluation homomorphisms are constructed from a chosen root α ∈ Fq of the polynomial f (x) that is at the heart of the PLWE instance. Then, this map is applied to every sample given and a number of distinguishability conditions giving rise to different attacks do appear. Each distinguishing condition defines a specific attack.  \nThe work of [8] studied the likelihood of some of the attacks presented in [5] . A statistical analysis showed that the distribution of the setting potentially v","cbCairRePbaXIczR","https://ap.wps.com/l/cbCairRePbaXIczR","pdf",520906,1,18,"English","en",105,"# Introduction\n## RLWE and PLWE as post-quantum paradigms\n## Structured vs unstructured lattice schemes\n## Existing frameworks for decisional attacks\n## Unbounded Small Values Attack and generalizations","[{\"question\":\"What is the main goal of the paper regarding root-based attacks on PLWE?\",\"answer\":\"To analyze whether root-based attacks can be extended using explicit ring isomorphisms constructed from vulnerable instances, and to determine whether this extension creates any new vulnerabilities.\"},{\"question\":\"Why does applying the constructed isomorphism not yield new vulnerabilities?\",\"answer\":\"Because the isomorphism distorts the PLWE samples in a way that the resulting samples cannot be used to distinguish, so the attack advantage does not carry over.\"},{\"question\":\"Does the paper restrict its results only to fully-split polynomial rings?\",\"answer\":\"The paper proves a strong characterization for fully-split rings and then extends the argument beyond fully-split settings, showing that the same distortion effect occurs regardless of the degree of the field extension.\"}]",1784188998,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"extensions-of-root-based-attacks-against-plwe-via-isomorphisms-of-rings","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/extensions-of-root-based-attacks-against-plwe-via-isomorphisms-of-rings/83582/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of the paper regarding root-based attacks on PLWE?","Question",{"text":75,"@type":76},"To analyze whether root-based attacks can be extended using explicit ring isomorphisms constructed from vulnerable instances, and to determine whether this extension creates any new vulnerabilities.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does applying the constructed isomorphism not yield new vulnerabilities?",{"text":80,"@type":76},"Because the isomorphism distorts the PLWE samples in a way that the resulting samples cannot be used to distinguish, so the attack advantage does not carry over.",{"name":82,"@type":73,"acceptedAnswer":83},"Does the paper restrict its results only to fully-split polynomial rings?",{"text":84,"@type":76},"The paper proves a strong characterization for fully-split rings and then extends the argument beyond fully-split settings, showing that the same distortion effect occurs regardless of the degree of the field 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