[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83711-en":3,"doc-seo-83711-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83711,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Derivative-Free Richelot Isogenies via Subresultants with Algebraic Certification","The classical Richelot (2,2)-isogeny for genus-2 curves builds a codomain triple (U, V, W) from a factorization f=uvw using Wronskian derivatives. A completely derivative-free reformulation over prime fields F_p with p>2 reconstructs the Wronskian output via 2×2 minors, obtained from first subresultants and a linear syzygy. The Remainder-Polynomial Route produces the exact polynomial identity in F_p[x]. The Guarded Subresultant Route adds constant-size algebraic guards and a lightweight post-check, with O(1) field operations per step and certified correctness validated by extensive randomized testing.","arXiv :2607 .03376v1 [math .NT] 3 Jul 2026  \nDerivative-Free Richelot Isogenies via Subresultants with Algebraic Certification  \nHung T. Danga,1,∗, Diep V. Nguyena  \na Department of Mathematics, University of Phuong Dong, Hanoi, Vietnam  \nAbstract  \nThe classical Richelot (2 , 2)-isogeny step for genus-2 curves constructs a codomain triple (U, V, W ) from a factorization f = uvw via Wronskian derivatives. We give a completely derivative-free reformulation over prime fields Fp , p > 2, by expressing the Wronskian output through the 2 × 2 minors of the coefficient matrix and recovering them from first subresultants and alinear syzygy. The resulting Remainder-Polynomial Route (RPR) is proven to produce the identical output triple in Fp [x] not merely up to units, but asan exact polynomial identity. Building on this equivalence, we introduce the Guarded Subresultant Route (GSR), a deterministic evaluator that certifies admissibility through constant-size algebraic guards, a lightweight post-check, and at most one bounded affine retry. All routes execute O(1) field operations per step. A prototype over 106 matched trials per prime confirms a 4.75–6× kernel speedup for RPR over the classical Wronskian formula, and the full GSR pipeline remains 1.4–3× faster than WRO despite the certification overhead. Correctness is independently verified by a double-Richelot involution test on 2.5 × 105 random triples across five primes.  \nKeywords: Richelot isogeny, subresultant, finite field arithmetic, polynomial remainder sequence, genus-2 curve, deterministic certification  \n∗ Corresponding author.  \nEmail address: [hung.dt@phuongdong.edu.vn](hung.dt@phuongdong.edu.vn) (Hung T. Dang)  \n1 ORCID: 0009-0006-3272-0573  \n1. Introduction  \nThe Richelot (2 , 2)-isogeny is the canonical gateway to explicit genus-2 isogenies: starting from a square-free sextic f = uvw with monic quadraticsu, v, w ∈ Fp [x], one obtains a codomain C′ : y 2 = UV W by algebraic relations among the factors. The classical route forms the Wronskian minors  \nU = v′ w − vw′, V = w′ u − wu′, W = u′ v − uv′,  \na construction that is elegant and well understood over odd characteristic. Yet it relies on polynomial differentiation and offers no intrinsic mechanism to certify that the output triple (U, V, W ) satisfies the admissibility conditions guaranteeing a smooth, separable codomain.  \nThese limitations become relevant in settings that demand deterministic, certifiable implementations of the (2 , 2)-step, such as hash functions built from Richelot isogeny chains [1] and related isogeny-based protocols [2 , 3] . This motivates two design goals: first, to remove differentiation from the critical path while preserving the exact algebraic output of the classical step; second, to provide a local, lightweight certificate enforcing the standard soundness criterion (pairwise coprime quadratics with nonzero discriminants), rather than assuming it or detecting failure only downstream.  \nApproach. We work within the subresultant framework and interpret the Richelot step in terms of the 2 × 2 minors: in degree 2, the three Plücker coordinates of the coefficient matrix encode the Wronskian output polynomial directly, and the first subresultant delivers two of the three coordinates via a single pseudo-remainder. This viewpoint naturally suggests a derivative-free reconstruction and an explicit way to check admissibility locally.  \nContributions. Our contribution is primarily algebraic and algorithmic in nature: we provide a subresultant-based reinterpretation of the Richelot step over finite fields, together with a certified evaluator enforcing the classical admissibility conditions.  \n1. Algebraic equality without derivatives. We recast the Richelot (2 , 2) step via the 2 × 2 minors (Plücker coordinates) of the coefficient matrix. The first subresultant encodes two of the three minors, anda linear syzygy relation recovers the third; assembling them yields the remainder-polynomial route (RPR","cbCaipL8kNOeInw4","https://ap.wps.com/l/cbCaipL8kNOeInw4","pdf",612912,2,1,30,"English","en",105,"# Introduction\n## Approach and Design Goals\n## Contributions\n## Positioning\n## Scope and Terminology\n## Paper Organization","[{\"question\":\"What problem does the paper address with the classical Richelot (2,2)-isogeny step?\",\"answer\":\"The classical construction relies on polynomial differentiation and does not provide an intrinsic way to certify that the produced (U,V,W) triple satisfies the admissibility conditions for a smooth, separable genus-2 codomain.\"},{\"question\":\"How does the Remainder-Polynomial Route (RPR) remove derivatives while preserving correctness?\",\"answer\":\"RPR reinterprets the Richelot step using 2×2 minors (Plücker coordinates) of the coefficient matrix, recovering needed minors through first subresultants and a linear syzygy. It is proven to match the classical Wronskian output exactly in F_p[x], not just up to units.\"},{\"question\":\"What does the Guarded Subresultant Route (GSR) add to ensure admissibility?\",\"answer\":\"GSR inserts constant-size algebraic guards (including discriminants and resultants related to subresultants) plus a lightweight post-check, allowing a single bounded affine retry. The route either returns a certified (U,V,W) triple or rejects non-admissible instances with deterministic control flow.\"}]",1784189906,76,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"derivative-free-richelot-isogenies-via-subresultants-with-algebraic-certification","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/derivative-free-richelot-isogenies-via-subresultants-with-algebraic-certification/83711/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","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 problem does the paper address with the classical Richelot (2,2)-isogeny step?","Question",{"text":75,"@type":76},"The classical construction relies on polynomial differentiation and does not provide an intrinsic way to certify that the produced (U,V,W) triple satisfies the admissibility conditions for a smooth, separable genus-2 codomain.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the Remainder-Polynomial Route (RPR) remove derivatives while preserving correctness?",{"text":80,"@type":76},"RPR reinterprets the Richelot step using 2×2 minors (Plücker coordinates) of the coefficient matrix, recovering needed minors through first subresultants and a linear syzygy. It is proven to match the classical Wronskian output exactly in F_p[x], not just up to units.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the Guarded Subresultant Route (GSR) add to ensure admissibility?",{"text":84,"@type":76},"GSR inserts constant-size algebraic guards (including discriminants and resultants related to subresultants) plus a lightweight post-check, allowing a single bounded affine retry. The route either returns a certified (U,V,W) triple or rejects non-admissible instances with deterministic control flow.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":22,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]