Construction of Permutation Polynomials over Finite Fields with the help of SCR polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913499033108480 |
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| author | Sharma, Bidushi Basnet, Dhiren Kumar |
| author_facet | Sharma, Bidushi Basnet, Dhiren Kumar |
| contents | In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over $\mathbb{F}_{q^{2}}$. The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in $μ_{q+1}$ (the set of $(q+1)-\emph{th}$ roots of unity), which helps in the determination of polynomials that permute $\mathbb{F}_{q^{2}}$. In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type $ax^{q+1}+bx^{q}+bx+a^{q}$ taking both the cases under consideration viz. $a\in \mathbb{F}_{q}$ and $a\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ both. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of Permutation Polynomials over Finite Fields with the help of SCR polynomials Sharma, Bidushi Basnet, Dhiren Kumar Number Theory 11T06, 11T55 In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over $\mathbb{F}_{q^{2}}$. The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in $μ_{q+1}$ (the set of $(q+1)-\emph{th}$ roots of unity), which helps in the determination of polynomials that permute $\mathbb{F}_{q^{2}}$. In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type $ax^{q+1}+bx^{q}+bx+a^{q}$ taking both the cases under consideration viz. $a\in \mathbb{F}_{q}$ and $a\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ both. |
| title | Construction of Permutation Polynomials over Finite Fields with the help of SCR polynomials |
| topic | Number Theory 11T06, 11T55 |
| url | https://arxiv.org/abs/2404.00927 |