Index Calculus in the Trace Zero Variety

TL;DR

This paper adapts Gaudry's index calculus algorithm to solve discrete logarithms in the trace zero variety of elliptic curves, focusing on degree 3 and 5 extensions, with theoretical and practical insights.

Contribution

It extends index calculus methods to trace zero varieties of elliptic curves, providing both theoretical analysis and prototype implementation results.

Findings

Effective discrete log solving in trace zero varieties for degree 3 and 5 extensions

Theoretical comparison with existing algorithms shows advantages in certain cases

Prototype implementation demonstrates practical feasibility

Abstract

We discuss how to apply Gaudry's index calculus algorithm for abelian varieties to solve the discrete logarithm problem in the trace zero variety of an elliptic curve. We treat in particular the practically relevant cases of field extensions of degree 3 or 5. Our theoretical analysis is compared to other algorithms present in the literature, and is complemented by results from a prototype implementation.