Some new results on permutation trinomials over finite fields with even characteristic
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908801230176256 |
|---|---|
| author | Garg, Kirpa Hasan, Sartaj Ul Vishwakarma, Chandan Kumar |
| author_facet | Garg, Kirpa Hasan, Sartaj Ul Vishwakarma, Chandan Kumar |
| contents | The construction of permutation trinomials of the form $X^r(X^{α(2^m-1)}+X^{β(2^m-1)} + 1)$ over $\F_{2^{2m}}$, where $m,~r\text{ and }α> β$ are positive integers, is an active area of research. Several classes of permutation trinomials with fixed values of $α$, $β$ and $r$ have been studied. Here, we construct three new classes of permutation trinomials with $(α,β,r)\in\{(7,5,7),(8,6,9),(10,4,11)\}$ over $\F_{2^{2m}}$. We also analyze the quasi-multiplicative equivalence of the newly obtained classes of permutation trinomials to both the existing ones and to each other. Additionally, we prove the nonexistence of a class of permutation trinomials over $\F_{2^{2m}}$ of the same type for $r=9$, $α=7$, and $β=3$ when $m > 3$. Furthermore, we provide a proof for a conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (2024). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some new results on permutation trinomials over finite fields with even characteristic Garg, Kirpa Hasan, Sartaj Ul Vishwakarma, Chandan Kumar Number Theory The construction of permutation trinomials of the form $X^r(X^{α(2^m-1)}+X^{β(2^m-1)} + 1)$ over $\F_{2^{2m}}$, where $m,~r\text{ and }α> β$ are positive integers, is an active area of research. Several classes of permutation trinomials with fixed values of $α$, $β$ and $r$ have been studied. Here, we construct three new classes of permutation trinomials with $(α,β,r)\in\{(7,5,7),(8,6,9),(10,4,11)\}$ over $\F_{2^{2m}}$. We also analyze the quasi-multiplicative equivalence of the newly obtained classes of permutation trinomials to both the existing ones and to each other. Additionally, we prove the nonexistence of a class of permutation trinomials over $\F_{2^{2m}}$ of the same type for $r=9$, $α=7$, and $β=3$ when $m > 3$. Furthermore, we provide a proof for a conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (2024). |
| title | Some new results on permutation trinomials over finite fields with even characteristic |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.14674 |