[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81760-en":3,"doc-seo-81760-105":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},81760,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Fast Deterministic Normal Bases and Circulant Polynomial Determinants","Deterministic construction of normal elements in finite field extensions E=Fq[x]/(Γ) of degree n is addressed, with Γ monic and irreducible. The work builds on the fact that for t∈Fq, βt=(θ−t)−1 is normal except for at most n(n−1) exceptional parameters. Normality is characterized via a “cleared Moore” circulant matrix whose determinant links to trace- and Gram-based formulations over Fq. A near-linear-time circulant polynomial determinant algorithm is developed, then extended to smaller base sizes using low-degree embeddings.","arXiv :2607 .003 13v 1 [ cs . SC] 1 Jul 2026  \nFast Deterministic Normal Bases and Circulant Polynomial Determinants  \nMark Giesbrecht \\# 􀀚 University of Waterloo, Canada Armin Jampshidpey \\# 􀀚 University of Waterloo, Canada Éric Schost \\# 􀀚  \nUniversity of Waterloo, Canada  \n~~ Abstract ~~  \nLet E = Fq [x]/(Γ) be an algebraic extension of degree n over the finite field Fq , given by a Γ ∈ Fq [x] monic and irreducible. It is classical that any such E contains an element β ∈ E that is normal over Fq , i.e., the conjugates β, β q , . . . , β qn − 1 form an Fq-basis of E over Fq . In this pa˜per we give a  \ndeterministic algorithm which finds such a normal element using Oϵ ((n2 log q)1+ϵ) + O (n log2 q) bit operations, for any ϵ > 0.  \nThe algorithm works by showing that, for a parameter t ∈ Fq , the element βt = (θ − t)−1 is normal except for at most n (n − 1) values of t. This is established by constructing a “cleared Moore”circulant matrix over Fqn [T], whose determinant degree at most n (n − 1), such that βt is normal if and only the determinant is non-zero at t ∈ Fq . For faster computation over the base field, wereplace this by an equivalent trace Gram circulant matrix over Fq [T] .  \nA main algorithmic contribution is a fast determinant algorithm for circulant matrices of polynomials, which uses triangular set projection and modular composition techniques to achieve a near-linear cost. Given an n × n circulant matrix over Fq [t] whose entries have degree at most m > 0 , we show how to compute its determinant deterministically with Oϵ ((nm log q)1+ϵ) bit operations.  \nWe complete the solution by showing how to extend this to finite fields of size less than n (n − 1), through an embedding in a low-degree extension field, at poly-logarithmic additional cost.  \n2012 ACM Subject Classification Mathematics of computing → Computations in finite fields; Computing methodologies → Algebraic algorithms; Theory of computation → Design and analysis of algorithms  \nKeywords and phrases Normal Basis, Finite Fields, Deterministic Algorithm, Circulant Determinant, Modular Composition  \nFunding Mark Giesbrecht: Natural Sciences and Engineering Council, Canada  \nÉric Schost: Natural Sciences and Engineering Council, Canada  \nJune 29, 2026 . Submitted for publication.  \n2 Fast Deterministic Normal Bases and Circulant Polynomial Determinants  \n1 Introduction  \nLet Fq be the finite field with q elements. A finite extension field E = Fq [x]/(Γ)  Fqn is constructed via a monic irreducible polynomial Γ ∈ Fq [x] of degree n. A normal basis of E/Fq is a Fq-basis of the form  \n{β, βq , . . . , β qn − 1 } ⊆ E  \nfor some normal element β ∈ E. It is an important classical result that a normal element β always exists. Our goal here is to construct such an element deterministically in nearly quadratic time with respect to n.  \nNormal bases have a number of computational advantages over other bases, such asthe power basis for finite field arithmetic. The most fundamental is that the qth-power Frobenius automorphism acts as a cyclic shift on the coordinates of any element represented in a normal basis, making the exponentiation by power of q essentially free. This makes normal bases attractive in cryptographic applications over binary fields, where they underpin efficient hardware multipliers for elliptic curve cryptography [21] . The same Frobenius shift property makes them useful in algorithms for factoring polynomials over finite fields, where manipulating the Frobenius map efficiently is central [12, 16] . Normal bases are also a key ingredient in algorithms for computing isomorphisms and embeddings between finite fields. The deterministic polynomial-time algorithm [20] constructs a normal basis as an essential subroutine, and subsequent work [6] has continued to rely on this approach. Normal bases further arise in the study of linear-feedback shift registers, in the construction of optimal codes, and in algorithms for computing Frobenius forms of matri","cbCaipul2EPdkLqP","https://ap.wps.com/l/cbCaipul2EPdkLqP","pdf",593996,3,1,15,"English","en",105,"# Introduction\n## Background on normal bases and Frobenius shifts\n## Deterministic vs randomized approaches\n## Output size considerations","[{\"question\":\"What problem does the paper solve?\",\"answer\":\"It gives a deterministic method to find a normal element in a finite field extension E/Fq defined by an irreducible polynomial Γ.\"},{\"question\":\"How does the algorithm decide when an element is normal?\",\"answer\":\"For βt=(θ−t)−1, normality is characterized by the non-vanishing of the determinant of an associated circulant matrix over a polynomial ring, which holds for all but at most n(n−1) values of t.\"},{\"question\":\"What key computational technique enables fast running time?\",\"answer\":\"A fast determinant algorithm for circulant matrices of polynomials is combined with techniques such as triangular set projection and modular composition, and then lifted via low-degree embeddings when the base field size is smaller.\"}]",1784175880,38,{"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},"fast-deterministic-normal-bases-and-circulant-polynomial-determinants","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/fast-deterministic-normal-bases-and-circulant-polynomial-determinants/81760/",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 solve?","Question",{"text":75,"@type":76},"It gives a deterministic method to find a normal element in a finite field extension E/Fq defined by an irreducible polynomial Γ.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the algorithm decide when an element is normal?",{"text":80,"@type":76},"For βt=(θ−t)−1, normality is characterized by the non-vanishing of the determinant of an associated circulant matrix over a polynomial ring, which holds for all but at most n(n−1) values of t.",{"name":82,"@type":73,"acceptedAnswer":83},"What key computational technique enables fast running time?",{"text":84,"@type":76},"A fast determinant algorithm for circulant matrices of polynomials is combined with techniques such as triangular set projection and modular composition, and then lifted via low-degree embeddings when the base field size is 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