An optimal representation for the trace zero subgroup

TL;DR

This paper introduces an optimal-size, efficient representation for elements of the trace zero subgroup in elliptic and hyperelliptic curves, facilitating compression, decompression, and scalar multiplication across various field extensions.

Contribution

It presents a new rational function-based representation for trace zero subgroup elements, optimized for any genus and prime degree extension, with practical algorithms and implementation insights.

Findings

Achieves minimal representation size for trace zero subgroup elements

Provides efficient compression and decompression algorithms

Demonstrates effectiveness in small genus and extension cases

Abstract

We give an optimal-size representation for the elements of the trace zero subgroup of the Picard group of an elliptic or hyperelliptic curve of any genus, with respect to a field extension of any prime degree. The representation is via the coefficients of a rational function, and it is compatible with scalar multiplication of points. We provide efficient compression and decompression algorithms, and complement them with implementation results. We discuss in detail the practically relevant cases of small genus and extension degree, and compare with the other known compression methods.