# Montgomery curves and their arithmetic

## TL;DR

This paper surveys the theory and cryptographic applications of Montgomery elliptic curves, focusing on their arithmetic, algorithms, and use in cryptosystems over non-binary finite fields.

## Contribution

It provides a comprehensive overview of Montgomery curves, including their x-only arithmetic, Ladder algorithm, Diffie-Hellman, and differential addition chains, highlighting their foundational role in cryptography.

## Key findings

- Detailed explanation of Montgomery's x-only arithmetic and Ladder algorithm
- Analysis of y-coordinate recovery and differential addition chains
- Survey of cryptographic protocols using Montgomery curves

## Abstract

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery's x-only arithmetic and Ladder algorithm, x-only Diffie--Hellman, y-coordinate recovery, and 2-dimensional and Euclidean differential addition chains such as Montgomery's PRAC algorithm.