Finite Fields and Their Applications最新文献

{"title":"Representations of group actions and their applications in cryptography","authors":"Giuseppe D'Alconzo,&nbsp;Antonio J. Di Scala","doi":"10.1016/j.ffa.2024.102476","DOIUrl":"10.1016/j.ffa.2024.102476","url":null,"abstract":"<div><p>Cryptographic group actions provide a flexible framework that allows the instantiation of several primitives, ranging from key exchange protocols to PRFs and digital signatures. The security of such constructions is based on the intractability of some computational problems. For example, given the group action <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo>,</mo><mo>⋆</mo><mo>)</mo></math></span>, the weak unpredictability assumption (Alamati et al. (2020) <span><span>[1]</span></span>) requires that, given random <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s in <em>X</em>, no probabilistic polynomial time algorithm can compute, on input <span><math><msub><mrow><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>g</mi><mo>⋆</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>Q</mi></mrow></msub></math></span> and <em>y</em>, the set element <span><math><mi>g</mi><mo>⋆</mo><mi>y</mi></math></span>.</p><p>In this work, we study such assumptions, aided by the definition of <em>group action representations</em> and a new metric, the <em>q-linear dimension</em>, that estimates the “linearity” of a group action, or in other words, how much it is far from being linear. We show that under some hypotheses on the group action representation, and if the <em>q</em>-linear dimension is polynomial in the security parameter, then the weak unpredictability and other related assumptions cannot hold. This technique is applied to some actions from cryptography, like the ones arising from the equivalence of linear codes, as a result, we obtain the impossibility of using such actions for the instantiation of certain primitives.</p><p>As an additional result, some bounds on the <em>q</em>-linear dimension are given for classical groups, such as <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>GL</mi><mo>(</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and the cyclic group <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> acting on itself.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102476"},"PeriodicalIF":1.2,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001151/pdfft?md5=da2ac4d07e20b23f31147c448a4a4dc4&pid=1-s2.0-S1071579724001151-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the stopping time of the Collatz map in F2[x]","authors":"Gil Alon ,&nbsp;Angelot Behajaina ,&nbsp;Elad Paran","doi":"10.1016/j.ffa.2024.102473","DOIUrl":"10.1016/j.ffa.2024.102473","url":null,"abstract":"<div><p>We study the stopping time of the Collatz map for a polynomial <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, and bound it by <span><math><mi>O</mi><mo>(</mo><mrow><mi>deg</mi></mrow><msup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mn>1.5</mn></mrow></msup><mo>)</mo></math></span>, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102473"},"PeriodicalIF":1.2,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959410","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 algebraic degrees of inverted Kloosterman sums","authors":"Xin Lin ,&nbsp;Daqing Wan","doi":"10.1016/j.ffa.2024.102477","DOIUrl":"10.1016/j.ffa.2024.102477","url":null,"abstract":"<div><p>The study of <em>n</em>-dimensional inverted Kloosterman sums was suggested by Katz (1995) <span><span>[7]</span></span> who handled the case when <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> from complex point of view. For general <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>n</em>-dimensional inverted Kloosterman sums were studied from both complex and <em>p</em>-adic point of view in our previous paper (2024) <span><span>[10]</span></span>. In this note, we study the algebraic degree of the inverted <em>n</em>-dimensional Kloosterman sum as an algebraic integer.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102477"},"PeriodicalIF":1.2,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959411","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":"Two classes of LCD BCH codes over finite fields","authors":"Yuqing Fu ,&nbsp;Hongwei Liu","doi":"10.1016/j.ffa.2024.102478","DOIUrl":"10.1016/j.ffa.2024.102478","url":null,"abstract":"<div><p>BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> and <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> with designed distance <span><math><mi>δ</mi><mo>=</mo><mi>ℓ</mi><msup><mrow><mi>q</mi></mrow><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>1</mn></math></span> are determined, where <span><math><mi>q</mi><mo>&gt;</mo><mn>2</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span>. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span> are developed for odd <em>q</em>, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span> being dually-BCH codes are presented, where <em>q</em> is odd and <span><math><mi>m</mi><mo>≢</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102478"},"PeriodicalIF":1.2,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950539","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":"New results on PcN and APcN polynomials over finite fields","authors":"Zhengbang Zha ,&nbsp;Lei Hu","doi":"10.1016/j.ffa.2024.102471","DOIUrl":"10.1016/j.ffa.2024.102471","url":null,"abstract":"<div><p>Permutation polynomials with low <em>c</em>-differential uniformity have important applications in cryptography and combinatorial design. In this paper, we investigate perfect <em>c</em>-nonlinear (PcN) and almost perfect <em>c</em>-nonlinear (APcN) polynomials over finite fields. Based on some known permutation polynomials, we present several classes of PcN or APcN polynomials by using the Akbary-Ghioca-Wang criterion.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102471"},"PeriodicalIF":1.2,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959583","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":"New results on n-to-1 mappings over finite fields","authors":"Xiaoer Qin ,&nbsp;Li Yan","doi":"10.1016/j.ffa.2024.102469","DOIUrl":"10.1016/j.ffa.2024.102469","url":null,"abstract":"<div><p><em>n</em>-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of <em>n</em>-to-1 mappings over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. Then we characterize a new form of AGW-like criterion, and use it to present many classes of <em>n</em>-to-1 polynomials with the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>h</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. Finally, by using monomials on the cosets of a subgroup of <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and another form of AGW-like criterion, we show some <em>n</em>-to-1 trinomials over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102469"},"PeriodicalIF":1.2,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141638722","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":"Circularity in finite fields and solutions of the equations xm + ym − zm = 1","authors":"Wen-Fong Ke ,&nbsp;Hubert Kiechle","doi":"10.1016/j.ffa.2024.102467","DOIUrl":"10.1016/j.ffa.2024.102467","url":null,"abstract":"<div><p>An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102467"},"PeriodicalIF":1.2,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141623614","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":"Probabilistic Galois theory in function fields","authors":"Alexei Entin,&nbsp;Alexander Popov","doi":"10.1016/j.ffa.2024.102466","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102466","url":null,"abstract":"<div><p>We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo><mo>[</mo><mi>y</mi><mo>]</mo></math></span> with i.i.d. coefficients <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> taking values in the set <span><math><mo>{</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo><mo>:</mo><mi>deg</mi><mo>⁡</mo><mi>a</mi><mo>≤</mo><mi>d</mi><mo>}</mo></math></span> with uniform probability, is irreducible with probability tending to <span><math><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></mfrac></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>, where <em>d</em> and <em>q</em> are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, then the Galois group of this polynomial is actually equal to the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with probability tending to <span><math><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></mfrac></math></span>. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with <em>n</em> fixed and <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102466"},"PeriodicalIF":1.2,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606199","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":"More classes of permutation pentanomials over finite fields with characteristic two","authors":"Tongliang Zhang ,&nbsp;Lijing Zheng ,&nbsp;Hanbing Zhao","doi":"10.1016/j.ffa.2024.102468","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102468","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we investigate permutation pentanomials over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;gcd&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We transform the problem concerning permutation property of &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; into demonstrating that the corresponding fractional polynomial permutes the unit circle &lt;em&gt;U&lt;/em&gt; of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with order &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; via a well-known lemma, and then into showing that there are no certain solution in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/ms","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102468"},"PeriodicalIF":1.2,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606200","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 most symmetric smooth cubic surface over a finite field of characteristic 2","authors":"Anastasia V. Vikulova","doi":"10.1016/j.ffa.2024.102470","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102470","url":null,"abstract":"<div><p>In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic 2. We prove that if the order of the field is a power of 4, then the automorphism group of maximal order of a smooth cubic surface over this field is <span><math><msub><mrow><mi>PSU</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. If the order of the field of characteristic 2 is not a power of 4, then we prove that the automorphism group of maximal order of a smooth cubic surface over this field is the symmetric group of degree 6. Moreover, we prove that smooth cubic surfaces with such properties are unique up to isomorphism.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102470"},"PeriodicalIF":1.2,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606206","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}