[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83910-en":3,"doc-seo-83910-105":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},83910,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Algebraic Modelings of the Supersingular Isogeny Problem","A new algebraic modeling of the Supersingular Isogeny Problem expresses supersingular elliptic curve connections as a system of multivariate polynomial equations, focusing on isogenies of degree 2^k or 3^k. The construction uses Renes formulas for elliptic curves in Montgomery form (degree 2) or triangular form (degree 3). The work proves the resulting systems are zero-dimensional, studies the dimension of their highest-degree components, and shows they are not in generic coordinates. Experiments indicate Gröbner-basis solving is faster than modular-polynomial modeling.","arXiv :2607 .05 160v 1 [ cs . SC] 6 Jul 2026  \nALGEBRAIC MODELINGS OF THE SUPERSINGULAR ISOGENY PROBLEM  \nALESSIO CAMINATA, ANDREA SANGUINETI, AND SILVIA SCONZA  \nAbstract . We present a new algebraic modeling of the Supersingular Isogeny Problem as a system of multivariate polynomial equations, in the case where the elliptic curves are connected by an isogeny whose degree is a power of 2 or 3. This modeling relies on Renes formulas for elliptic curvesin Montgomery form (degree 2) or triangular form (degree 3) . We investigate several algebraic properties of these systems: we prove that they are zero-dimensional, compute the dimension of their highest degree part, and show that they are not in generic coordinates. Experimental results show that solving these systems via Gröbner basis techniques is significantly faster than solving the algebraic modeling with modular polynomials.  \nIntroduction  \nWe fix a prime number p > 3 and let E and E′ be two supersingular elliptic curves defined over Fp2 such that there exists a degree d isogeny φ : E → E′. The Supersingular Isogeny Problem (SIP) asks to find φ . When d and p are sufficiently large, a random instance of this problem is believed to be computationally difficult to solve, even using a quantum computer. For this reason, it has been used as the underlying problem for several post-quantum cryptographic schemes, such as the signature scheme SQIsign [20], which, at the time of writing, is admitted to Round 3 of the NIST call for post-quantum signature schemes. For this reason, the SIP has been widely studied and several algorithms and methods have been proposed in the literature. We note that, since the degree of isogenies is multiplicative under composition, the main difficulty in solving the SIP arises when the degree d is a power of a prime ℓ  p. In this setting, the most efficient approaches exploit structural properties of the graph of ℓ-isogenies between supersingular elliptic curves [15 , 17 , 21 , 26] .  \nGiven the growing importance of post-quantum cryptography, both the cryptanalysis of proposed schemes and the study of the hardness of their underlying problems have received increasing attention in recent years. A comprehensive security assessment requires considering attacks originating from different areas of post-quantum cryptography. In this context, algebraic modeling and algebraic attacks have emerged as important tools. In the literature, several algebraic models have been proposed and studied for problems arising in code-based cryptography [11 , 19 , 32 , 38] and lattice-based cryptography [2 , 3 , 42] . In contrast, algebraic modeling of problems in isogeny-based cryptography has received comparatively less attention [44] .  \nA natural approach to modeling the Supersingular Isogeny Problem as a system of multivariate polynomial equations is via modular polynomials [48, §10] . Given a positive integer N  p the N-th modular polynomial ΦN(X, Y ) is a polynomial (with integer coefficients) with the property that ΦN(j1 , j2 ) = 0 if and only if j 1 , j2 are the j-invariants of elliptic curves that are related by an isogeny of degree N. So, if N = ℓe, where ℓ is a prime number, given the j-invariantsjstart, jfinish of two elliptic curves Estart, Efinish connected by a N-isogeny φ, we can reduce the SIP to several smaller instances of the SIP in degree ℓ by solving a multivariate polynomial system (see Polynomial system 1) with modular polynomials, whose solutions represent all possible paths from  \nKey words and phrases. Supersingular Isogeny Problem, Gröbner bases, solving degree, algebraic modeling.  \n2 ALESSIO CAMINATA, ANDREA SANGUINETI, AND SILVIA SCONZA  \nEstart to Efinish in the ℓ-isogeny graph. This approach has been studied in the paper [44] . There, the authors propose the algebraic model and study the complexity of solving the corresponding polynomial system via Gröbner basis techniques.  \nIn this paper, we propose two distinct algebraic models for the SIP","cbCaiqGyLgZrJ3tv","https://ap.wps.com/l/cbCaiqGyLgZrJ3tv","pdf",682386,4,1,24,"English","en",105,"# Introduction\n## Background on the Supersingular Isogeny Problem\n## Modular-polynomial based modeling\n## Proposed algebraic models (degrees 2^k and 3^k)\n## Properties of the resulting polynomial systems\n## Experimental comparison of solving approaches","[{\"question\":\"What is the Supersingular Isogeny Problem (SIP) addressed in the document?\",\"answer\":\"Given supersingular elliptic curves E and E′ over Fp2 with an isogeny φ : E → E′ of degree d, the SIP asks to determine φ.\"},{\"question\":\"How does the paper model the SIP algebraically?\",\"answer\":\"It reduces connections between elliptic curves to a system of multivariate polynomial equations, built using Renes formulas to derive Renes polynomials analogous to modular polynomials.\"},{\"question\":\"Which degrees of isogenies are covered, and what does the approach rely on?\",\"answer\":\"The modeling covers isogenies whose degrees are powers of 2 or 3, using Montgomery-form formulas for degree 2 and triangular-form formulas for degree 3.\"}]",1784191388,60,{"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},"algebraic-modelings-of-the-supersingular-isogeny-problem","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/algebraic-modelings-of-the-supersingular-isogeny-problem/83910/",{"url":52,"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","2026-07-24","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 Supersingular Isogeny Problem (SIP) addressed in the document?","Question",{"text":75,"@type":76},"Given supersingular elliptic curves E and E′ over Fp2 with an isogeny φ : E → E′ of degree d, the SIP asks to determine φ.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper model the SIP algebraically?",{"text":80,"@type":76},"It reduces connections between elliptic curves to a system of multivariate polynomial equations, built using Renes formulas to derive Renes polynomials analogous to modular polynomials.",{"name":82,"@type":73,"acceptedAnswer":83},"Which degrees of isogenies are covered, and what does the approach rely on?",{"text":84,"@type":76},"The modeling covers isogenies whose degrees are powers of 2 or 3, using Montgomery-form formulas for degree 2 and triangular-form formulas for degree 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