Large class of many-to-one mappings over quadratic extension of finite fields
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915189473935360 |
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| author | Zheng, Yanbin Zhang, Meiying Ding, Yanjin Zha, Zhengbang Wang, Qiang |
| author_facet | Zheng, Yanbin Zhang, Meiying Ding, Yanjin Zha, Zhengbang Wang, Qiang |
| contents | Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b x + c) + u x^q + v x$, where $h(x) \in \mathbb{F}_{q^2}[x]$ and $a$, $b$, $c$, $u$, $v \in \mathbb{F}_{q^2}$. Using a commutative diagram satisfied by $f(x)$ and trace functions over finite fields, we reduce the problem whether $f(x)$ is a many-to-one mapping on $\mathbb{F}_{q^2}$ to another problem whether an associated polynomial $g(x)$ is a many-to-one mapping on the subfield $\mathbb{F}_{q}$. In particular, when $h(x) = x^{r}$ and $r$ satisfies certain conditions, we reduce $g(x)$ to polynomials of small degree or linearized polynomials. Then by employing the many-to-one properties of these low degree or linearized polynomials on $\mathbb{F}_{q}$, we derive a series of explicit characterization for $f(x)$ to be many-to-one on $\mathbb{F}_{q^2}$. On the other hand, for all $1$-to-$1$ mappings obtained in this paper, we determine the inverses of these permutation polynomials. Moreover, we also explicitly construct involutions from $2$-to-$1$ mappings of this form. Our findings generalize and unify many results in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large class of many-to-one mappings over quadratic extension of finite fields Zheng, Yanbin Zhang, Meiying Ding, Yanjin Zha, Zhengbang Wang, Qiang Information Theory Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b x + c) + u x^q + v x$, where $h(x) \in \mathbb{F}_{q^2}[x]$ and $a$, $b$, $c$, $u$, $v \in \mathbb{F}_{q^2}$. Using a commutative diagram satisfied by $f(x)$ and trace functions over finite fields, we reduce the problem whether $f(x)$ is a many-to-one mapping on $\mathbb{F}_{q^2}$ to another problem whether an associated polynomial $g(x)$ is a many-to-one mapping on the subfield $\mathbb{F}_{q}$. In particular, when $h(x) = x^{r}$ and $r$ satisfies certain conditions, we reduce $g(x)$ to polynomials of small degree or linearized polynomials. Then by employing the many-to-one properties of these low degree or linearized polynomials on $\mathbb{F}_{q}$, we derive a series of explicit characterization for $f(x)$ to be many-to-one on $\mathbb{F}_{q^2}$. On the other hand, for all $1$-to-$1$ mappings obtained in this paper, we determine the inverses of these permutation polynomials. Moreover, we also explicitly construct involutions from $2$-to-$1$ mappings of this form. Our findings generalize and unify many results in the literature. |
| title | Large class of many-to-one mappings over quadratic extension of finite fields |
| topic | Information Theory |
| url | https://arxiv.org/abs/2503.06640 |