{"title":"Constructing permutation polynomials from permutation polynomials of subfields","authors":"Lucas Reis , Qiang Wang","doi":"10.1016/j.ffa.2024.102415","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we study the permutational property of polynomials of the form <span><math><mi>f</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>k</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>⋅</mo><mi>M</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>L</mi><mo>,</mo><mi>M</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> are <em>q</em>-linearized polynomials and <span><math><mi>k</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> satisfies a generic condition. We specialize in the case where <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the linearized <em>q</em>-associate of <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>a</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>−</mo><mi>a</mi><mo>)</mo></math></span>, <em>t</em> is a divisor of <em>n</em> and <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> satisfies <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi><mo>/</mo><mi>t</mi></mrow></msup><mo>=</mo><mn>1</mn></math></span>. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> from permutations of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub></math></span>, by simply solving a system of independent equations of the form <span><math><msub><mrow><mi>Tr</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub><mo>(</mo><msup><mrow><mi>δ</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, where the <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are the coefficients of <em>f</em>. In fact, the same method can be applied to construct complete mappings of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> from complete mappings of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102415"},"PeriodicalIF":1.2000,"publicationDate":"2024-03-29","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/S1071579724000546","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
In this paper we study the permutational property of polynomials of the form over the finite field , where are q-linearized polynomials and satisfies a generic condition. We specialize in the case where is the linearized q-associate of , t is a divisor of n and satisfies . This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of from permutations of , by simply solving a system of independent equations of the form , where the 's are the coefficients of f. In fact, the same method can be applied to construct complete mappings of from complete mappings of .
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.