Bounds on Heights of $2$-isogeny Graphs in Ordinary Curves over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$ and Its Application

TL;DR

This paper improves bounds on the height of 2-isogeny volcano graphs for ordinary elliptic curves over finite fields, providing tighter estimates especially over quadratic extensions, with practical bounds for prime fields.

Contribution

It introduces improved bounds on the height of 2-isogeny volcano graphs over finite fields, including a tighter bound for quadratic extensions and a method to compute bounds for prime fields.

Findings

Bound h < log2(√4q) for 2-volcano graphs.

Tighter bound h ≤ ⌊(1/2)⌊log2 p⌋⌋ + 2 for q = p^2.

Method to compute bounds for q = p.

Abstract

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.