Finite Fields and Their Applications最新文献

{"title":"Simple matrix cryptosystem is not broken by Liu's attack","authors":"Lih-Chung Wang,&nbsp;Yen-Liang Kuan,&nbsp;Po-En Tseng,&nbsp;Chun-Yen Chou","doi":"10.1016/j.ffa.2025.102643","DOIUrl":"10.1016/j.ffa.2025.102643","url":null,"abstract":"<div><div>At PQCrypto2013, Tao et al. proposed a new multivariate public key cryptosystem for encryption called Simple Matrix (or ABC) encryption scheme. In 2018, Liu et al. proposed a key recovery attack on ABC scheme with claimed complexity of <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>s</mi></mrow><mrow><mn>4</mn><mi>w</mi></mrow></msup><mo>)</mo></mrow></math></span>, where <em>s</em> is the size of the <span><math><mi>s</mi><mo>×</mo><mi>s</mi></math></span> square matrices in the scheme, <span><math><mi>w</mi><mo>=</mo><mn>3</mn></math></span> in the usual Gaussian elimination algorithm and <span><math><mi>w</mi><mo>=</mo><mn>2.3776</mn></math></span> in improved scheme. In this paper, we show that Liu's attack only works for the case <span><math><mi>s</mi><mo>=</mo><mn>2</mn></math></span> of ABC scheme which means that Liu's attack doesn't break ABC scheme.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102643"},"PeriodicalIF":1.2,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143928788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Several new classes of p-ary weakly regular plateaued functions and minimal codes with several weights","authors":"Wengang Jin,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2025.102644","DOIUrl":"10.1016/j.ffa.2025.102644","url":null,"abstract":"<div><div>Plateaued functions, including bent functions, are crucial in cryptography due to their possession of a range of desirable cryptographic properties. Weakly regular plateaued functions can also be employed in many domains. In particular, they have been widely used in designing good linear codes for several applications (such as secret sharing and two-party computation), association schemes, and strongly regular graphs. This paper is devoted to weakly regular plateaued functions, whose objectives are twofold. First, we aim to generate new infinite families of weakly regular plateaued functions and then, to design new families of <em>p</em>-ary linear codes and investigate their use for some standard applications after studying its minimality based on their weight distributions. More specifically, we present several classes of weakly regular plateaued functions from monomial bent functions, and determine their corresponding dual functions explicitly. Furthermore, we exploit our constructions to derive several new classes of minimal linear codes violating the Ashikhmin-Barg condition with six, seven, nine, ten or eleven weights, which are more appropriate for several applications.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102644"},"PeriodicalIF":1.2,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A tighter bound on the minimum distances for an infinite family of binary BCH codes and its generalization","authors":"Haodong Lu,&nbsp;Xuan Wang,&nbsp;Minjia Shi","doi":"10.1016/j.ffa.2025.102628","DOIUrl":"10.1016/j.ffa.2025.102628","url":null,"abstract":"<div><div>In this paper, we improve the bound on the minimum distance for the family of binary cyclic codes proposed by Sun et al. (2024) <span><span>[8]</span></span>. The 3-ary analogue is also studied in this paper, which is a nice family of ternary cyclic codes that contains some best known linear codes, and this family has a better lower bound on minimum distance than that of codes proposed by Chen et al. (2023) <span><span>[2]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102628"},"PeriodicalIF":1.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Products of involutions in symplectic groups over general fields (II): Finite fields","authors":"Clément de Seguins Pazzis","doi":"10.1016/j.ffa.2025.102641","DOIUrl":"10.1016/j.ffa.2025.102641","url":null,"abstract":"<div><div>Let <em>s</em> be an <em>n</em>-dimensional symplectic form over a field <span><math><mi>F</mi></math></span> of characteristic other than 2, with <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math></span>.</div><div>In a previous article, we have proved that if <span><math><mi>F</mi></math></span> is infinite then every element of the symplectic group <span><math><mi>Sp</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> is the product of four involutions if <em>n</em> is a multiple of 4 and of five involutions otherwise.</div><div>Here, we adapt this result to all finite fields with characteristic not 2, with the sole exception of the very special situation where <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span> and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>=</mo><mn>3</mn></math></span>, a special case which we study extensively.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102641"},"PeriodicalIF":1.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the index of power compositional polynomials","authors":"Sumandeep Kaur ,&nbsp;Surender Kumar ,&nbsp;László Remete","doi":"10.1016/j.ffa.2025.102642","DOIUrl":"10.1016/j.ffa.2025.102642","url":null,"abstract":"<div><div>The index of a monic irreducible polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> having a root <em>θ</em> is the index <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo></math></span> where <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the ring of algebraic integers of the number field <span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math></span>. If <span><math><mo>[</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>:</mo><mi>Z</mi><mo>[</mo><mi>θ</mi><mo>]</mo><mo>]</mo><mo>=</mo><mn>1</mn></math></span>, then <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> belonging to <span><math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, to be monogenic. As an application of our results, for a polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>A</mi><mo>⋅</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, with <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>deg</mi><mo>⁡</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>&lt;</mo><mi>d</mi></math></span> and <span><math><mo>|</mo><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, we prove that for each positive integer <em>k</em> with <span><math><mi>rad</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>|</mo><mi>rad</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, the power compositional polynomial <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is monogenic if and only if <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is monogenic, provided that <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> is irreducible. At the end of the paper, we give infinite families of polynomials as examples.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102642"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a theorem of Kara and Klyachko","authors":"Sanmin Wang","doi":"10.1016/j.ffa.2025.102645","DOIUrl":"10.1016/j.ffa.2025.102645","url":null,"abstract":"<div><div>There exists a finite set of pairs <span><math><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of points of the modular curve <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> with <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> but <span><math><mo>(</mo><mi>j</mi><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mi>j</mi><mo>(</mo><mi>N</mi><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>=</mo><mo>(</mo><mi>j</mi><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>,</mo><mi>j</mi><mo>(</mo><mi>N</mi><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo></math></span>, which are the singularities of the plane model <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>. A theorem of Kara and Klyachko gives a characterization of these pairs of points of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>. However, their proof for this theorem contains an erroneous assertion. Following their idea, we give an elementary proof for this theorem in details.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102645"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The second generalized covering radius of binary primitive double-error-correcting BCH codes","authors":"Lev Yohananov ,&nbsp;Moshe Schwartz","doi":"10.1016/j.ffa.2025.102648","DOIUrl":"10.1016/j.ffa.2025.102648","url":null,"abstract":"<div><div>We completely determine the second covering radius for binary primitive double-error-correcting BCH codes. As part of this process, we provide a lower bound on the second covering radius for binary primitive BCH codes correcting more than two errors.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102648"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the b-symbol weights of linear codes for large b","authors":"Dongmei Huang ,&nbsp;Qunying Liao ,&nbsp;Gaohua Tang ,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102647","DOIUrl":"10.1016/j.ffa.2025.102647","url":null,"abstract":"<div><div>Recently, <em>b</em>-symbol metric linear codes, which are extensions of linear codes with Hamming weight and symbol-pair weight, have attracted much attention due to their repair efficiency in <em>b</em>-symbol read channels. Extensive efforts have been dedicated to studying the classic Hamming weight and symbol-pair weight of linear codes, but there has been relatively less progress regarding the general <em>b</em>-symbol weight for <span><math><mi>b</mi><mo>≥</mo><mn>3</mn></math></span>. In this paper, we mainly consider the <em>b</em>-symbol weight of linear codes for large <em>b</em>. Firstly, we establish a direct correspondence between the <em>b</em>-symbol weight distribution of a linear code and the Hamming weight distribution of another linear code. Using this relation, we derive a Griesmer-type bound on the minimum <em>b</em>-symbol weight, which partially settles a conjecture of Shi, Zhu and Helleseth. Furthermore, we discover that the minimum <em>b</em>-symbol distance of a <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> cyclic code is equal to the minimum <em>b</em>-symbol distance of certain linear code of parameters <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>−</mo><mi>b</mi><mo>]</mo></math></span> and prove that every <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> cyclic code is always <em>b</em>-symbol MDS for <span><math><mi>b</mi><mo>≥</mo><mi>min</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>)</mo></mrow></math></span>, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is the minimum distance of its dual. When <span><math><mi>b</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo>⊥</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math></span> we also determine its <em>b</em>-symbol weight distribution. Finally, we investigate the minimum <em>b</em>-symbol weight of several specific cyclic codes, including QR codes, irreducible cyclic codes, Kasami codes and the duals of double-error-correcting BCH codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102647"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some counterexamples for a version of Springer's theorem on forms of higher degree","authors":"Alexander S. Sivatski","doi":"10.1016/j.ffa.2025.102646","DOIUrl":"10.1016/j.ffa.2025.102646","url":null,"abstract":"<div><div>Let <em>φ</em> be a form (homogeneous polynomial) of degree <em>d</em> in <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> variables over a field <em>F</em>. We call <em>φ</em> anisotropic over <em>F</em>, if the equality <span><math><mi>φ</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></math></span> with <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>F</mi></math></span> implies <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>…</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span>. Otherwise <em>φ</em> is called isotropic. Assume that <span><math><mi>L</mi><mo>/</mo><mi>F</mi></math></span> be a finite field extension, the numbers <em>d</em> and <span><math><mo>[</mo><mi>L</mi><mo>:</mo><mi>F</mi><mo>]</mo></math></span> are coprime, and the form <em>φ</em> is anisotropic and diagonal. Does the form <em>φ</em> remain anisotropic over <em>L</em>? This problem can be considered as an analog of the Springer theorem on behaviour of anisotropic quadratic forms under odd degree extensions. In particular, we investigate this problem in the case <span><math><mi>F</mi><mo>=</mo><mi>Q</mi></math></span>. We give examples of extensions of degree 2, 3, and 5, which show that in general the answer is negative and pose a few related questions. Cubic extensions are treated in the arithmetic as well as in the general case. Finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and their extensions is the principal tool in constructing the counterexamples in question.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"107 ","pages":"Article 102646"},"PeriodicalIF":1.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The automorphism group of the pn-torsion points of an elliptic curve over a field of characteristic p ≥ 5","authors":"Bo-Hae Im ,&nbsp;Hansol Kim","doi":"10.1016/j.ffa.2025.102631","DOIUrl":"10.1016/j.ffa.2025.102631","url":null,"abstract":"<div><div>For a field <em>K</em> of characteristic <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span> and the elliptic curve <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>s</mi><mi>x</mi><mo>+</mo><mi>t</mi></math></span> defined over the function field <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> of two variables <em>s</em> and <em>t</em>, we prove that for a positive integer <em>n</em>, the automorphism group of the normal extension <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow><mo>)</mo></mrow><mo>/</mo><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is isomorphic to <span><math><msup><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mi>Z</mi><mo>)</mo></mrow><mrow><mo>×</mo></mrow></msup></math></span>, and its inseparable degree is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102631"},"PeriodicalIF":1.2,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}