Bounds on Heights of $2$-isogeny Graphs in Ordinary Curves over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$ and Its Application
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909302292217856 |
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| author | Hashimoto, Yuji Nuida, Koji |
| author_facet | Hashimoto, Yuji Nuida, Koji |
| contents | It is known that any isogeny graph consisting of ordinary elliptic curves over $\mathbb{F}_q$ with $q = p$ or $p^2$ has a special structure, called a volcano graph. We have a bound $h < \log_2 \sqrt{4q}$ of a height $h$ of the $2$-volcano graph. In this paper, we improve the bound on a height of $2$-volcano graphs over $\mathbb{F}_q$. In case $q = p^2$, we show a tighter bound $h \leq \left\lfloor \frac{ 1 }{ 2 } \lfloor \log_2 p \rfloor \right\rfloor + 2 $. In case $q = p$, we also show that a good bound for each prime $p$ can be computed by using our proposed techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00505 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on Heights of $2$-isogeny Graphs in Ordinary Curves over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$ and Its Application Hashimoto, Yuji Nuida, Koji Number Theory It is known that any isogeny graph consisting of ordinary elliptic curves over $\mathbb{F}_q$ with $q = p$ or $p^2$ has a special structure, called a volcano graph. We have a bound $h < \log_2 \sqrt{4q}$ of a height $h$ of the $2$-volcano graph. In this paper, we improve the bound on a height of $2$-volcano graphs over $\mathbb{F}_q$. In case $q = p^2$, we show a tighter bound $h \leq \left\lfloor \frac{ 1 }{ 2 } \lfloor \log_2 p \rfloor \right\rfloor + 2 $. In case $q = p$, we also show that a good bound for each prime $p$ can be computed by using our proposed techniques. |
| title | Bounds on Heights of $2$-isogeny Graphs in Ordinary Curves over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$ and Its Application |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.00505 |