Index Calculus Attack for Hyperelliptic Curves of Small Genus

Abstract

We present a variation of the index calculus attack by Gaudry which can be used to solve the discrete logarithm problem in the Jacobian of hyperelliptic curves. The new algorithm has a running time which is better than the original index calculus attack and the Rho method (and other square-root algorithms) for curves of genus ≥ 3. We also describe another improvement for curves of genus ≥ 4 (slightly slower, but less dependent on memory space) initially mentioned by Harley and used in a number of papers, but never analyzed in details.