[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84572-en":3,"doc-seo-84572-105":29,"detail-sidebar-cat-0-en-105":90},{"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},84572,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Complexity of Low-Degree Skew Polynomial Multiplication over Finite Fields","Study of multiplication complexity in skew polynomial rings over finite fields. The work proves that multiplying two elements in F_{q^n}[x;σ] of degree at most d\u003Cn can be carried out using O(ω_K/ K) up to a structured factor of size (dω_K−1n), yielding a quasi-optimal bound consistent with the lower bound of Chen–Ye (ISSAC’24). The reduction converts the finite-field setting to a split algebra via equivariant multiplication theory of Couveignes–Ezome (J. Algebra, 2023), then applies existing fast algorithms.","arXiv :2607 .00476v 1 [ cs . SC] 1 Jul 2026  \nComplexity of Low-Degree Skew Polynomial Multiplication over Finite Fields  \nKe Ye 1 , Yichuan Cao 1 , Ruichen Qiu 1  \nJuly 2, 2026  \nAbstract  \nIn this note, we study the complexity of multiplication in skew polynomial rings over finite fields. We prove that the product of two elements in F qn [x; σ] of degree at most d \u003C n can beautcompuomorptedhismusingThis(ωtK−1 n)ches thaerithconjmetic oecturalpuerationspper boouvnedroFqf C,awhruseor–eL iBsotrhegneq[FrobeSSACn’ i1u7s] and is quasi-optimal in view of the lower bound of Chen–Ye [ISSAC’24] . The proof reduces the finite-field case to the split algebra case using the equivariant multiplication theory of Couveignes–Ezome [J. Algebra, 2023], and then applies existing fast algorithms.  \n1 Introduction  \nSkew polynomial rings are rings of non-commutative polynomials introduced by Ore [13] . They appear naturally throughout computational and non-commutative algebra. For example, these non-commutative rings provide algebraic models for rank-metric and Gabidulin-type codes [4, 5, 11, 12], and interact with Gröbner bases [10] and structured matrix multiplication [8] . From the algorithmic point of view, multiplication is the basic operation on which division, factorization, interpolation, and coding-theoretic procedures depend. The first fast algorithm for skew polynomial multiplication was discovered by Giesbrecht in 1998 [6] . The known fastest algorithms include the general algorithms of Caruso–Le Borgne [1], sparse-support methods of Giesbrecht–Huang–Schost [7], and special low-degree algorithms of Chen–Ye [2] .  \nThroughout the paper, we use K for a field, and A/K for an étale K-algebra. We also assume that there is a K-linear automorphism σ of A such that Aσ = K, and the order of σ is equal to dim KA. Such a K-algebra is called a ⟨σ⟩-Galois algebra. Two typical examples are (Kn , τ ) and (L, σ), where Kn is the split K-algebra and τ is the cyclic left shift operator sending (a1 , . . . , an) to (a2 , . . . , an , a1 ) , L is a cyclic extension of K and σ ∈ Gal (L/K) is a generator.  \nThe underlying additive group of the skew polynomial ring A [x; σ] is the ordinary polynomial ring A [x] . Given A, B ∈ A[x; σ], we write f = Pi aixi , g = Pj bjxj for some ai , bj ∈ A. Then the product fg ∈ A[x; σ] is defined as fg = Pk ckxk , where  \nck := X aiσi(bj) . (1)  \ni+j =k  \nGiven a positive integer d, we denote by A[x; σ]≤d the subspace of A[x; σ] consisting of all skew polynomials of degree at most d. We consider the map  \nµ d : A[x; σ]≤d× A[x; σ]≤d → A[x; σ]≤2d , (f , g) →7 fg.  \nLet CK(µ d) be the computational complexity of µ d. Concerning the value of CK(µ d), we have the following conjecture.  \nConjecture 1. [1] For any ⟨σ⟩-Galois algebra A/K and positive integer d, we have CK(µ d) = 􀀀pldimitionn(,vr)ωKK− 2 􀀁 where n = dim KA and ωK is the exponential of the complexity of the matrix mul-  \n1 State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences. This work is supported by the Strategic Priority Research Program of Chinese Academy of Sciences under Grant XDA0480502 and XDA0480503 .  \nAccording to [1], Conjecture 1 is known to be true for d ≥ n. Thus, it suffices to assume d \u003C n. Moreover, it is proved in [2] that CK(µ d) is bounded below by Ω􀀀dn min(d, n)ωK −2􀀁, showing that any algorithm for µ d achieving the conjectured upper bound is quasi-optimal. The goal of this note is to prove Conjecture 1 for the case where A/K = Fqn/Fq .  \nTheorem 2. Let q be a prime power and let n be a positive integer. If σ is the Fro benius generator of Gal (Fqn/Fq), then for any 0 \u003C d \u003C n, we have  \nCK(µ d) = (dωK −1n) .  \nThe main results presented in this paper are obtained through an interaction between the authors and an artificial intelligence agent system, MechMath Agent Team (MMAT) [9] . The authors assume full responsibility for the paper","cbCaioqse2htR33l","https://ap.wps.com/l/cbCaioqse2htR33l","pdf",293710,1,4,"English","en",105,"# Abstract\n# Introduction\n## Background and motivation\n## Setup and definitions\n## Conjecture and goal\n# Preliminaries\n# Proof of Theorem 2","[{\"question\":\"What multiplication problem is analyzed in this note?\",\"answer\":\"The note analyzes the computational complexity of multiplying skew polynomials in rings of the form F_{q^n}[x;σ] with bounded degree.\"},{\"question\":\"Under what degree condition do the main results apply?\",\"answer\":\"The main statement targets the regime 0\\u003cd\\u003cn, i.e., multiplying elements of degree at most d strictly less than n.\"},{\"question\":\"How is the finite-field multiplication reduced to a more tractable case?\",\"answer\":\"The proof reduces the finite-field case to the split-algebra case using equivariant multiplication theory, then leverages existing fast algorithms for the split setting.\"}]",1784196870,10,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"complexity-of-low-degree-skew-polynomial-multiplication-over-finite-fields","",{"@graph":35,"@context":84},[36,52,67],{"@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":21},"https://docshare.wps.com/document/complexity-of-low-degree-skew-polynomial-multiplication-over-finite-fields/84572/",{"url":51,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What multiplication problem is analyzed in this note?","Question",{"text":74,"@type":75},"The note analyzes the computational complexity of multiplying skew polynomials in rings of the form F_{q^n}[x;σ] with bounded degree.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Under what degree condition do the main results apply?",{"text":79,"@type":75},"The main statement targets the regime 0\u003Cd\u003Cn, i.e., multiplying elements of degree at most d strictly less than n.",{"name":81,"@type":72,"acceptedAnswer":82},"How is the finite-field multiplication reduced to a more tractable case?",{"text":83,"@type":75},"The proof reduces the finite-field case to the split-algebra case using equivariant multiplication theory, then leverages existing fast algorithms for the split setting.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":28,"slug":132},"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]