{"title":"New constructions of permutation polynomials of the form x+γTrqq2(h(x)) over finite fields with even characteristic","authors":"Sha Jiang, Mu Yuan, Kangquan Li, Longjiang Qu","doi":"10.1016/j.ffa.2024.102522","DOIUrl":null,"url":null,"abstract":"<div><div>Permutation polynomials over finite fields are widely used in cryptography, coding theory, and combinatorial design. Particularly, permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> have been studied by many researchers and applied to lift minimal blocking sets. In this paper, we further investigate permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> over finite fields with even characteristic. On the one hand, guided by the idea of choosing functions <em>h</em> with a low <em>q</em>-degree, we completely determine the sufficient and necessary conditions of <em>γ</em> for six classes of polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> with <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>2</mn></mrow></msup></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> (<span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>4</mn></math></span>) to be permutations. These results determine the sizes of directions of these six functions, which is generally difficult. On the other hand, we slightly generalize the above idea and construct other six classes of permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> with <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> (<span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>4</mn></math></span>). We believe that more results about permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> can be obtained by exploiting this idea.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102522"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-23","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/S1071579724001618","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Permutation polynomials over finite fields are widely used in cryptography, coding theory, and combinatorial design. Particularly, permutation polynomials of the form have been studied by many researchers and applied to lift minimal blocking sets. In this paper, we further investigate permutation polynomials of the form over finite fields with even characteristic. On the one hand, guided by the idea of choosing functions h with a low q-degree, we completely determine the sufficient and necessary conditions of γ for six classes of polynomials of the form with and () to be permutations. These results determine the sizes of directions of these six functions, which is generally difficult. On the other hand, we slightly generalize the above idea and construct other six classes of permutation polynomials of the form with and (). We believe that more results about permutation polynomials of the form can be obtained by exploiting this idea.
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.