Paper 2025/1568 Original: eprint.iacr.org/2025/1568 ↗

Montgomery Curves: Exact Enumeration and Probabilistic Analysis

Abstract

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.