Paper 2025/1568
Montgomery Curves: Exact Enumeration and Probabilistic Analysis
, Shanghai Jiao Tong UniversityAbstract
This paper addresses the question of how many distinct Montgomery curves exist over a finite field $\mathbb{F}_p$ for a given order. We provide a complete answer by presenting precise counting formulas from two perspectives: (1) the number of $\mathbb{F}_p$-isomorphism classes, and (2) the number of coefficient pairs $(A,B)$. Additionally, we propose conjectural formulas that approximate the probability of a randomly chosen curve having an order with a specific, cryptographically relevant structure. As a byproduct of our methods, we also establish a novel identity for the Kronecker-Hurwitz class number.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Preprint.
- Keywords
- Montgomery curvesElliptic curves
- Contact author(s)
- tsaiyiju @ sjtu edu cn
- History
- 2025-09-03: approved
- 2025-09-01: received
- See all versions
- Short URL
- https://ia.cr/2025/1568
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2025/1568,
author = {Tsai Yi-Ju},
title = {Montgomery Curves: Exact Enumeration and Probabilistic Analysis},
howpublished = {Cryptology {ePrint} Archive, Paper 2025/1568},
year = {2025},
url = {https://eprint.iacr.org/2025/1568}
}
