[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81520-en":3,"doc-seo-81520-105":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},81520,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Computing Isomorphisms Between Products of Supersingular Elliptic Curves","Deligne–Ogus–Shioda theorem ensures existence of isomorphisms between products of supersingular elliptic curves over finite fields, and this work makes them computable. An explicit algorithm is given to compute such isomorphisms from the curves’ endomorphism rings. Assuming GRH, the method runs in expected polynomial time. Using Deuring correspondence, isogeny computations become quaternion algebra tasks, reduced to integer systems of quadratic and linear equations from norm equations, solved via ℓ-adic techniques in a low-discriminant subring, yielding an efficient Las Vegas probabilistic algorithm.","arXiv :2503 .21535v2 [math .NT] 10 Jul 2026  \nCOMPUTING ISOMORPHISMS BETWEEN PRODUCTS OF SUPERSINGULAR ELLIPTIC CURVES  \nPIERRICK GAUDRY, JULIEN SOUMIER, PIERRE-JEAN SPAENLEHAUER  \nAbstract . The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields.  \nIn this paper, we present an algorithm for explicitly computing these isomorphisms given the endomorphism rings of the curves. Under GRH, it is proved to run in expected polynomial time. Our approach leverages the Deuring correspondence, enabling us to reformulate computational isogeny problems into algebraic problems in quaternions. Specifically, we reduce the computation of isomorphisms to solving systems of quadratic and linear equations over the integers derived from norm equations. We develop ℓ-adic techniques for solving these equations when we have access to a low discriminant subring. Combining these results leads to the description of an efficient probabilistic Las Vegas algorithm for computing the desired isomorphisms.  \n1. Introduction  \nAlgorithms for computing isogenies between elliptic curves have been a vast field of research, leading to the recent development of higher-dimensional techniques in cryptography [17, 8] . In particular, abelian varieties of dimension g ≥ 2 isomorphic to a product of supersingular elliptic curves play an important role in this setting [5, 18, 23, 3, 2] . An important feature of such abelian varieties is that they are all isomorphic over an algebraic closure. Let Fq be a finite field of characteristic p > 0. An abelian variety defined over Fq is superspecial  if it is Fq-isomorphic to a product of supersingular elliptic curves defined over Fq . The Deligne-Ogus-Shioda theorem [25, Thm. 3.5] states that for all g > 1, all dimension-g superspecial abelian varieties defined over Fq are Fq-isomorphic (as unpolarized abelian varieties) . The aim of this paper is to investigate computational aspects of this theorem.  \nProblem 1.1 (Effective Deligne-Ogus-Shioda problem) . Let g ≥ 2 be an integer. Given supersingular elliptic curves E1 ,..., Eg and E′1,..., E′g defined over Fq , compute an Fq-isomorphism E1 × · · · × Eg → E′1 × · · · × E′g .  \nThis appears to be a difficult computational problem. Indeed, computing the endomorphism ring of a supersingular curve is a computational problem which is considered hard, and the security of several cryptographic constructions relies on it [3, 2] . Solving Problem 1.1 would provide non-trivial information about the endomorphism rings of the curves: From an isomorphism E1 × E2 → E′1 × E′2, we caEPrn1ooemEmp1ui1ts1eifninoguetrnhieesrogacoleantneienxsotnφw-tijrhiver:EiaeljewoaEmve′io,reapxnhitrdsatmihneoffocoErmmp1atoIniosnit:tiohitnsh2pn1φerdo2,1er2spφth1ui1dysm: rings of the elliptic curves are given (see Section 2.2.2 for technical details on the  \n2  \nencoding of the endomorphism rings) . In this setting, Deuring’s correspondence allows us to translate Problem 1.1 into a problem about quaternion algebras.  \n1.1. Contributions. We focus on the case g = 2, which is the base case which serves as a building block for the general case g ≥ 2. Therefore our main problem is the computation of an isomorphism E1 × E2 → E′1 × E′2 between two products of supersingular elliptic curves, assuming that their endomorphisms rings are known. Endomorphism rings are given via an efficient representation of a Z-basis together with an explicit isomorphism with a maximal order in the quaternion algebra Bp,∞ . Our main contribution is a polynomial-time algorithm that computes an isomorphism E1 × E2 → E′1 × E′2 between products of maximal elliptic curves over Fp2 , assuming that we know the endomorphism rings of the curves. This algorithm relies on two main subroutines. The first one describes how to build a two by two matrix of isogenies which is an isomorphism, given its first column. The second one allows us to compute isomorphisms of the form","cbCaif5HcOccPq1g","https://ap.wps.com/l/cbCaif5HcOccPq1g","pdf",591429,3,1,21,"English","en",105,"# Introduction\n## Contributions\n## Related works","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper studies effective computation of an Fq-isomorphism between products of supersingular elliptic curves, starting from the given endomorphism rings.\"},{\"question\":\"How does the algorithm transform isomorphism computation into a more tractable task?\",\"answer\":\"It uses Deuring’s correspondence to reformulate computational isogeny problems as algebraic problems in quaternion algebras, reducing the task to solving systems derived from norm equations.\"},{\"question\":\"What complexity guarantee is proven for the proposed method?\",\"answer\":\"Under GRH, the algorithm is proved to run in expected polynomial time and is implemented as an efficient probabilistic Las Vegas algorithm for computing the desired 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problem does the paper address?","Question",{"text":75,"@type":76},"The paper studies effective computation of an Fq-isomorphism between products of supersingular elliptic curves, starting from the given endomorphism rings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the algorithm transform isomorphism computation into a more tractable task?",{"text":80,"@type":76},"It uses Deuring’s correspondence to reformulate computational isogeny problems as algebraic problems in quaternion algebras, reducing the task to solving systems derived from norm equations.",{"name":82,"@type":73,"acceptedAnswer":83},"What complexity guarantee is proven for the proposed method?",{"text":84,"@type":76},"Under GRH, the algorithm is proved to run in expected polynomial time and is implemented as an efficient probabilistic Las Vegas algorithm for computing the desired 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