{"title":"特性为 2 的有限域上的更多类置换五元数","authors":"Tongliang Zhang , Lijing Zheng , Hanbing Zhao","doi":"10.1016/j.ffa.2024.102468","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper, we investigate permutation pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup></math></span> with <span><math><mrow><mi>gcd</mi></mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msup><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We transform the problem concerning permutation property of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> into demonstrating that the corresponding fractional polynomial permutes the unit circle <em>U</em> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> with order <span><math><mi>q</mi><mo>+</mo><mn>1</mn></math></span> via a well-known lemma, and then into showing that there are no certain solution in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for some high-degree equations over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> associated with the fractional polynomial. According to numerical data, we have found all such permutations with <span><math><mn>4</mn><mo>≤</mo><mi>t</mi><mo><</mo><mn>100</mn><mo>,</mo><mn>1</mn><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mi>t</mi></math></span>, <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span>. Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle <em>U</em> found in this paper.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102468"},"PeriodicalIF":1.2000,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"More classes of permutation pentanomials over finite fields with characteristic two\",\"authors\":\"Tongliang Zhang , Lijing Zheng , Hanbing Zhao\",\"doi\":\"10.1016/j.ffa.2024.102468\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper, we investigate permutation pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup></math></span> with <span><math><mrow><mi>gcd</mi></mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msup><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We transform the problem concerning permutation property of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> into demonstrating that the corresponding fractional polynomial permutes the unit circle <em>U</em> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> with order <span><math><mi>q</mi><mo>+</mo><mn>1</mn></math></span> via a well-known lemma, and then into showing that there are no certain solution in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for some high-degree equations over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> associated with the fractional polynomial. According to numerical data, we have found all such permutations with <span><math><mn>4</mn><mo>≤</mo><mi>t</mi><mo><</mo><mn>100</mn><mo>,</mo><mn>1</mn><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mi>t</mi></math></span>, <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span>. Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle <em>U</em> found in this paper.</p></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"98 \",\"pages\":\"Article 102468\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-07-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724001072\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724001072","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
More classes of permutation pentanomials over finite fields with characteristic two
Let . In this paper, we investigate permutation pentanomials over of the form with . We transform the problem concerning permutation property of into demonstrating that the corresponding fractional polynomial permutes the unit circle U of with order via a well-known lemma, and then into showing that there are no certain solution in for some high-degree equations over associated with the fractional polynomial. According to numerical data, we have found all such permutations with , . Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle U found in this paper.
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.