Index Calculus in the Trace Zero Variety
This work addresses a cryptographic security problem for elliptic curve cryptography, but it is incremental as it adapts an existing algorithm to a specific variety.
The paper tackles the discrete logarithm problem in the trace zero variety of elliptic curves by applying Gaudry's index calculus algorithm, focusing on field extensions of degree 3 or 5, and provides theoretical analysis and prototype implementation results.
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.