Arithmetic of split Kummer surfaces: Montgomery endomorphism of Edwards products

TL;DR

This paper investigates the explicit arithmetic of split Kummer surfaces derived from elliptic curves, focusing on endomorphisms relevant for efficient scalar multiplication in cryptographic applications.

Contribution

It introduces new explicit formulas for the arithmetic of split Kummer surfaces and explores their endomorphisms, enhancing scalar multiplication methods.

Findings

Derived explicit arithmetic formulas for split Kummer surfaces.

Analyzed endomorphisms induced by elliptic curve addition.

Potential improvements for cryptographic scalar multiplication.

Abstract

Let $E$ be an elliptic curve, $\mathcal{K}_1$ its Kummer curve $E/\{\pm1\}$, $E^2$ its square product, and $\mathcal{K}_2$ the split Kummer surface $E^2/\{\pm1\}$. The addition law on $E^2$ gives a large endomorphism ring, which induce endomorphisms of $\mathcal{K}_2$. With a view to the practical applications to scalar multiplication on $\mathcal{K}_1$, we study the explicit arithmetic of $\mathcal{K}_2$.